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G = C7×C24.C22  order 448 = 26·7

Direct product of C7 and C24.C22

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C7×C24.C22, C2.7(D4×C28), C22⋊C4⋊5C28, (C2×C42)⋊2C14, C14.108(C4×D4), (C2×C28).358D4, C23.8(C2×C28), C24.5(C2×C14), C22.37(D4×C14), C2.C42⋊3C14, (C23×C14).2C22, C14.134(C4⋊D4), C14.64(C4.4D4), (C22×C28).33C22, C23.61(C22×C14), C22.37(C22×C28), C14.58(C42⋊C2), C14.33(C42⋊2C2), (C22×C14).452C23, C14.89(C22.D4), (C2×C4×C28)⋊3C2, (C2×C4⋊C4)⋊3C14, (C14×C4⋊C4)⋊30C2, C2.3(C7×C4⋊D4), (C7×C22⋊C4)⋊11C4, (C2×C4).34(C2×C28), (C2×C4).101(C7×D4), C2.2(C7×C4.4D4), (C2×C28).191(C2×C4), (C2×C14).604(C2×D4), (C2×C22⋊C4).6C14, C2.3(C7×C42⋊2C2), C22.22(C7×C4○D4), (C14×C22⋊C4).26C2, (C22×C4).88(C2×C14), (C22×C14).19(C2×C4), (C7×C2.C42)⋊5C2, C2.10(C7×C42⋊C2), (C2×C14).212(C4○D4), C2.5(C7×C22.D4), (C2×C14).224(C22×C4), SmallGroup(448,796)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C24.C22
Chief series
C1 — C2 — C22 — C23 — C22×C14 — C22×C28 — C14×C22⋊C4 — C7×C24.C22
Lower central
C1 — C22 — C7×C24.C22
Upper central
C1 — C22×C14 — C7×C24.C22

Generators and relations for C7×C24.C22
 G = < a,b,c,d,e,f,g | a7=b2=c2=d2=e2=1, f2=e, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fbf-1=bc=cb, gbg-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd, ef=fe, eg=ge >

Subgroups: 322 in 190 conjugacy classes, 90 normal (62 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C24, C28, C2×C14, C2×C14, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C24.C22, C4×C28, C7×C22⋊C4, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C23×C14, C7×C2.C42, C2×C4×C28, C14×C22⋊C4, C14×C4⋊C4, C7×C24.C22
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22×C4, C2×D4, C4○D4, C28, C2×C14, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C2×C28, C7×D4, C22×C14, C24.C22, C22×C28, D4×C14, C7×C4○D4, C7×C42⋊C2, D4×C28, C7×C4⋊D4, C7×C22.D4, C7×C4.4D4, C7×C42⋊2C2, C7×C24.C22

Smallest permutation representation of C7×C24.C22
►On 224 points
Generators in S224
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)(29 30 31 32 33 34 35)(36 37 38 39 40 41 42)(43 44 45 46 47 48 49)(50 51 52 53 54 55 56)(57 58 59 60 61 62 63)(64 65 66 67 68 69 70)(71 72 73 74 75 76 77)(78 79 80 81 82 83 84)(85 86 87 88 89 90 91)(92 93 94 95 96 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)(113 114 115 116 117 118 119)(120 121 122 123 124 125 126)(127 128 129 130 131 132 133)(134 135 136 137 138 139 140)(141 142 143 144 145 146 147)(148 149 150 151 152 153 154)(155 156 157 158 159 160 161)(162 163 164 165 166 167 168)(169 170 171 172 173 174 175)(176 177 178 179 180 181 182)(183 184 185 186 187 188 189)(190 191 192 193 194 195 196)(197 198 199 200 201 202 203)(204 205 206 207 208 209 210)(211 212 213 214 215 216 217)(218 219 220 221 222 223 224)

(1 77)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 33)(9 34)(10 35)(11 29)(12 30)(13 31)(14 32)(15 219)(16 220)(17 221)(18 222)(19 223)(20 224)(21 218)(22 46)(23 47)(24 48)(25 49)(26 43)(27 44)(28 45)(36 213)(37 214)(38 215)(39 216)(40 217)(41 211)(42 212)(50 59)(51 60)(52 61)(53 62)(54 63)(55 57)(56 58)(64 89)(65 90)(66 91)(67 85)(68 86)(69 87)(70 88)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(99 108)(100 109)(101 110)(102 111)(103 112)(104 106)(105 107)(113 140)(114 134)(115 135)(116 136)(117 137)(118 138)(119 139)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 142)(128 143)(129 144)(130 145)(131 146)(132 147)(133 141)(155 181)(156 182)(157 176)(158 177)(159 178)(160 179)(161 180)(162 207)(163 208)(164 209)(165 210)(166 204)(167 205)(168 206)(169 187)(170 188)(171 189)(172 183)(173 184)(174 185)(175 186)(190 203)(191 197)(192 198)(193 199)(194 200)(195 201)(196 202)

(1 59)(2 60)(3 61)(4 62)(5 63)(6 57)(7 58)(8 28)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 217)(16 211)(17 212)(18 213)(19 214)(20 215)(21 216)(29 48)(30 49)(31 43)(32 44)(33 45)(34 46)(35 47)(36 222)(37 223)(38 224)(39 218)(40 219)(41 220)(42 221)(50 77)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(64 83)(65 84)(66 78)(67 79)(68 80)(69 81)(70 82)(85 95)(86 96)(87 97)(88 98)(89 92)(90 93)(91 94)(99 143)(100 144)(101 145)(102 146)(103 147)(104 141)(105 142)(106 133)(107 127)(108 128)(109 129)(110 130)(111 131)(112 132)(113 149)(114 150)(115 151)(116 152)(117 153)(118 154)(119 148)(120 139)(121 140)(122 134)(123 135)(124 136)(125 137)(126 138)(155 199)(156 200)(157 201)(158 202)(159 203)(160 197)(161 198)(162 189)(163 183)(164 184)(165 185)(166 186)(167 187)(168 188)(169 205)(170 206)(171 207)(172 208)(173 209)(174 210)(175 204)(176 195)(177 196)(178 190)(179 191)(180 192)(181 193)(182 194)

(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 213)(9 214)(10 215)(11 216)(12 217)(13 211)(14 212)(15 25)(16 26)(17 27)(18 28)(19 22)(20 23)(21 24)(29 39)(30 40)(31 41)(32 42)(33 36)(34 37)(35 38)(43 220)(44 221)(45 222)(46 223)(47 224)(48 218)(49 219)(50 79)(51 80)(52 81)(53 82)(54 83)(55 84)(56 78)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 92)(64 74)(65 75)(66 76)(67 77)(68 71)(69 72)(70 73)(99 117)(100 118)(101 119)(102 113)(103 114)(104 115)(105 116)(106 135)(107 136)(108 137)(109 138)(110 139)(111 140)(112 134)(120 130)(121 131)(122 132)(123 133)(124 127)(125 128)(126 129)(141 151)(142 152)(143 153)(144 154)(145 148)(146 149)(147 150)(155 173)(156 174)(157 175)(158 169)(159 170)(160 171)(161 172)(162 191)(163 192)(164 193)(165 194)(166 195)(167 196)(168 190)(176 186)(177 187)(178 188)(179 189)(180 183)(181 184)(182 185)(197 207)(198 208)(199 209)(200 210)(201 204)(202 205)(203 206)

(1 77)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 222)(9 223)(10 224)(11 218)(12 219)(13 220)(14 221)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 29)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 36)(43 211)(44 212)(45 213)(46 214)(47 215)(48 216)(49 217)(50 59)(51 60)(52 61)(53 62)(54 63)(55 57)(56 58)(64 89)(65 90)(66 91)(67 85)(68 86)(69 87)(70 88)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(99 137)(100 138)(101 139)(102 140)(103 134)(104 135)(105 136)(106 115)(107 116)(108 117)(109 118)(110 119)(111 113)(112 114)(120 145)(121 146)(122 147)(123 141)(124 142)(125 143)(126 144)(127 152)(128 153)(129 154)(130 148)(131 149)(132 150)(133 151)(155 193)(156 194)(157 195)(158 196)(159 190)(160 191)(161 192)(162 171)(163 172)(164 173)(165 174)(166 175)(167 169)(168 170)(176 201)(177 202)(178 203)(179 197)(180 198)(181 199)(182 200)(183 208)(184 209)(185 210)(186 204)(187 205)(188 206)(189 207)

(1 189 77 207)(2 183 71 208)(3 184 72 209)(4 185 73 210)(5 186 74 204)(6 187 75 205)(7 188 76 206)(8 101 222 139)(9 102 223 140)(10 103 224 134)(11 104 218 135)(12 105 219 136)(13 99 220 137)(14 100 221 138)(15 152 30 127)(16 153 31 128)(17 154 32 129)(18 148 33 130)(19 149 34 131)(20 150 35 132)(21 151 29 133)(22 146 37 121)(23 147 38 122)(24 141 39 123)(25 142 40 124)(26 143 41 125)(27 144 42 126)(28 145 36 120)(43 108 211 117)(44 109 212 118)(45 110 213 119)(46 111 214 113)(47 112 215 114)(48 106 216 115)(49 107 217 116)(50 171 59 162)(51 172 60 163)(52 173 61 164)(53 174 62 165)(54 175 63 166)(55 169 57 167)(56 170 58 168)(64 201 89 176)(65 202 90 177)(66 203 91 178)(67 197 85 179)(68 198 86 180)(69 199 87 181)(70 200 88 182)(78 159 94 190)(79 160 95 191)(80 161 96 192)(81 155 97 193)(82 156 98 194)(83 157 92 195)(84 158 93 196)

(1 115 59 151)(2 116 60 152)(3 117 61 153)(4 118 62 154)(5 119 63 148)(6 113 57 149)(7 114 58 150)(8 175 28 204)(9 169 22 205)(10 170 23 206)(11 171 24 207)(12 172 25 208)(13 173 26 209)(14 174 27 210)(15 198 217 161)(16 199 211 155)(17 200 212 156)(18 201 213 157)(19 202 214 158)(20 203 215 159)(21 197 216 160)(29 179 48 191)(30 180 49 192)(31 181 43 193)(32 182 44 194)(33 176 45 195)(34 177 46 196)(35 178 47 190)(36 186 222 166)(37 187 223 167)(38 188 224 168)(39 189 218 162)(40 183 219 163)(41 184 220 164)(42 185 221 165)(50 133 77 106)(51 127 71 107)(52 128 72 108)(53 129 73 109)(54 130 74 110)(55 131 75 111)(56 132 76 112)(64 139 83 120)(65 140 84 121)(66 134 78 122)(67 135 79 123)(68 136 80 124)(69 137 81 125)(70 138 82 126)(85 104 95 141)(86 105 96 142)(87 99 97 143)(88 100 98 144)(89 101 92 145)(90 102 93 146)(91 103 94 147)
 
G:=sub<Sym(224)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,77)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,33)(9,34)(10,35)(11,29)(12,30)(13,31)(14,32)(15,219)(16,220)(17,221)(18,222)(19,223)(20,224)(21,218)(22,46)(23,47)(24,48)(25,49)(26,43)(27,44)(28,45)(36,213)(37,214)(38,215)(39,216)(40,217)(41,211)(42,212)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,89)(65,90)(66,91)(67,85)(68,86)(69,87)(70,88)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(99,108)(100,109)(101,110)(102,111)(103,112)(104,106)(105,107)(113,140)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,141)(155,181)(156,182)(157,176)(158,177)(159,178)(160,179)(161,180)(162,207)(163,208)(164,209)(165,210)(166,204)(167,205)(168,206)(169,187)(170,188)(171,189)(172,183)(173,184)(174,185)(175,186)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202), (1,59)(2,60)(3,61)(4,62)(5,63)(6,57)(7,58)(8,28)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,217)(16,211)(17,212)(18,213)(19,214)(20,215)(21,216)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(36,222)(37,223)(38,224)(39,218)(40,219)(41,220)(42,221)(50,77)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(64,83)(65,84)(66,78)(67,79)(68,80)(69,81)(70,82)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)(99,143)(100,144)(101,145)(102,146)(103,147)(104,141)(105,142)(106,133)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,149)(114,150)(115,151)(116,152)(117,153)(118,154)(119,148)(120,139)(121,140)(122,134)(123,135)(124,136)(125,137)(126,138)(155,199)(156,200)(157,201)(158,202)(159,203)(160,197)(161,198)(162,189)(163,183)(164,184)(165,185)(166,186)(167,187)(168,188)(169,205)(170,206)(171,207)(172,208)(173,209)(174,210)(175,204)(176,195)(177,196)(178,190)(179,191)(180,192)(181,193)(182,194), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,213)(9,214)(10,215)(11,216)(12,217)(13,211)(14,212)(15,25)(16,26)(17,27)(18,28)(19,22)(20,23)(21,24)(29,39)(30,40)(31,41)(32,42)(33,36)(34,37)(35,38)(43,220)(44,221)(45,222)(46,223)(47,224)(48,218)(49,219)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,78)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,92)(64,74)(65,75)(66,76)(67,77)(68,71)(69,72)(70,73)(99,117)(100,118)(101,119)(102,113)(103,114)(104,115)(105,116)(106,135)(107,136)(108,137)(109,138)(110,139)(111,140)(112,134)(120,130)(121,131)(122,132)(123,133)(124,127)(125,128)(126,129)(141,151)(142,152)(143,153)(144,154)(145,148)(146,149)(147,150)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,191)(163,192)(164,193)(165,194)(166,195)(167,196)(168,190)(176,186)(177,187)(178,188)(179,189)(180,183)(181,184)(182,185)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,77)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,222)(9,223)(10,224)(11,218)(12,219)(13,220)(14,221)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,29)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,36)(43,211)(44,212)(45,213)(46,214)(47,215)(48,216)(49,217)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,89)(65,90)(66,91)(67,85)(68,86)(69,87)(70,88)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(99,137)(100,138)(101,139)(102,140)(103,134)(104,135)(105,136)(106,115)(107,116)(108,117)(109,118)(110,119)(111,113)(112,114)(120,145)(121,146)(122,147)(123,141)(124,142)(125,143)(126,144)(127,152)(128,153)(129,154)(130,148)(131,149)(132,150)(133,151)(155,193)(156,194)(157,195)(158,196)(159,190)(160,191)(161,192)(162,171)(163,172)(164,173)(165,174)(166,175)(167,169)(168,170)(176,201)(177,202)(178,203)(179,197)(180,198)(181,199)(182,200)(183,208)(184,209)(185,210)(186,204)(187,205)(188,206)(189,207), (1,189,77,207)(2,183,71,208)(3,184,72,209)(4,185,73,210)(5,186,74,204)(6,187,75,205)(7,188,76,206)(8,101,222,139)(9,102,223,140)(10,103,224,134)(11,104,218,135)(12,105,219,136)(13,99,220,137)(14,100,221,138)(15,152,30,127)(16,153,31,128)(17,154,32,129)(18,148,33,130)(19,149,34,131)(20,150,35,132)(21,151,29,133)(22,146,37,121)(23,147,38,122)(24,141,39,123)(25,142,40,124)(26,143,41,125)(27,144,42,126)(28,145,36,120)(43,108,211,117)(44,109,212,118)(45,110,213,119)(46,111,214,113)(47,112,215,114)(48,106,216,115)(49,107,217,116)(50,171,59,162)(51,172,60,163)(52,173,61,164)(53,174,62,165)(54,175,63,166)(55,169,57,167)(56,170,58,168)(64,201,89,176)(65,202,90,177)(66,203,91,178)(67,197,85,179)(68,198,86,180)(69,199,87,181)(70,200,88,182)(78,159,94,190)(79,160,95,191)(80,161,96,192)(81,155,97,193)(82,156,98,194)(83,157,92,195)(84,158,93,196), (1,115,59,151)(2,116,60,152)(3,117,61,153)(4,118,62,154)(5,119,63,148)(6,113,57,149)(7,114,58,150)(8,175,28,204)(9,169,22,205)(10,170,23,206)(11,171,24,207)(12,172,25,208)(13,173,26,209)(14,174,27,210)(15,198,217,161)(16,199,211,155)(17,200,212,156)(18,201,213,157)(19,202,214,158)(20,203,215,159)(21,197,216,160)(29,179,48,191)(30,180,49,192)(31,181,43,193)(32,182,44,194)(33,176,45,195)(34,177,46,196)(35,178,47,190)(36,186,222,166)(37,187,223,167)(38,188,224,168)(39,189,218,162)(40,183,219,163)(41,184,220,164)(42,185,221,165)(50,133,77,106)(51,127,71,107)(52,128,72,108)(53,129,73,109)(54,130,74,110)(55,131,75,111)(56,132,76,112)(64,139,83,120)(65,140,84,121)(66,134,78,122)(67,135,79,123)(68,136,80,124)(69,137,81,125)(70,138,82,126)(85,104,95,141)(86,105,96,142)(87,99,97,143)(88,100,98,144)(89,101,92,145)(90,102,93,146)(91,103,94,147)>;
 
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,77)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,33)(9,34)(10,35)(11,29)(12,30)(13,31)(14,32)(15,219)(16,220)(17,221)(18,222)(19,223)(20,224)(21,218)(22,46)(23,47)(24,48)(25,49)(26,43)(27,44)(28,45)(36,213)(37,214)(38,215)(39,216)(40,217)(41,211)(42,212)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,89)(65,90)(66,91)(67,85)(68,86)(69,87)(70,88)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(99,108)(100,109)(101,110)(102,111)(103,112)(104,106)(105,107)(113,140)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,142)(128,143)(129,144)(130,145)(131,146)(132,147)(133,141)(155,181)(156,182)(157,176)(158,177)(159,178)(160,179)(161,180)(162,207)(163,208)(164,209)(165,210)(166,204)(167,205)(168,206)(169,187)(170,188)(171,189)(172,183)(173,184)(174,185)(175,186)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202), (1,59)(2,60)(3,61)(4,62)(5,63)(6,57)(7,58)(8,28)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,217)(16,211)(17,212)(18,213)(19,214)(20,215)(21,216)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(36,222)(37,223)(38,224)(39,218)(40,219)(41,220)(42,221)(50,77)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(64,83)(65,84)(66,78)(67,79)(68,80)(69,81)(70,82)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)(99,143)(100,144)(101,145)(102,146)(103,147)(104,141)(105,142)(106,133)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,149)(114,150)(115,151)(116,152)(117,153)(118,154)(119,148)(120,139)(121,140)(122,134)(123,135)(124,136)(125,137)(126,138)(155,199)(156,200)(157,201)(158,202)(159,203)(160,197)(161,198)(162,189)(163,183)(164,184)(165,185)(166,186)(167,187)(168,188)(169,205)(170,206)(171,207)(172,208)(173,209)(174,210)(175,204)(176,195)(177,196)(178,190)(179,191)(180,192)(181,193)(182,194), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,213)(9,214)(10,215)(11,216)(12,217)(13,211)(14,212)(15,25)(16,26)(17,27)(18,28)(19,22)(20,23)(21,24)(29,39)(30,40)(31,41)(32,42)(33,36)(34,37)(35,38)(43,220)(44,221)(45,222)(46,223)(47,224)(48,218)(49,219)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,78)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,92)(64,74)(65,75)(66,76)(67,77)(68,71)(69,72)(70,73)(99,117)(100,118)(101,119)(102,113)(103,114)(104,115)(105,116)(106,135)(107,136)(108,137)(109,138)(110,139)(111,140)(112,134)(120,130)(121,131)(122,132)(123,133)(124,127)(125,128)(126,129)(141,151)(142,152)(143,153)(144,154)(145,148)(146,149)(147,150)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,191)(163,192)(164,193)(165,194)(166,195)(167,196)(168,190)(176,186)(177,187)(178,188)(179,189)(180,183)(181,184)(182,185)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,77)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,222)(9,223)(10,224)(11,218)(12,219)(13,220)(14,221)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,29)(22,37)(23,38)(24,39)(25,40)(26,41)(27,42)(28,36)(43,211)(44,212)(45,213)(46,214)(47,215)(48,216)(49,217)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,89)(65,90)(66,91)(67,85)(68,86)(69,87)(70,88)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(99,137)(100,138)(101,139)(102,140)(103,134)(104,135)(105,136)(106,115)(107,116)(108,117)(109,118)(110,119)(111,113)(112,114)(120,145)(121,146)(122,147)(123,141)(124,142)(125,143)(126,144)(127,152)(128,153)(129,154)(130,148)(131,149)(132,150)(133,151)(155,193)(156,194)(157,195)(158,196)(159,190)(160,191)(161,192)(162,171)(163,172)(164,173)(165,174)(166,175)(167,169)(168,170)(176,201)(177,202)(178,203)(179,197)(180,198)(181,199)(182,200)(183,208)(184,209)(185,210)(186,204)(187,205)(188,206)(189,207), (1,189,77,207)(2,183,71,208)(3,184,72,209)(4,185,73,210)(5,186,74,204)(6,187,75,205)(7,188,76,206)(8,101,222,139)(9,102,223,140)(10,103,224,134)(11,104,218,135)(12,105,219,136)(13,99,220,137)(14,100,221,138)(15,152,30,127)(16,153,31,128)(17,154,32,129)(18,148,33,130)(19,149,34,131)(20,150,35,132)(21,151,29,133)(22,146,37,121)(23,147,38,122)(24,141,39,123)(25,142,40,124)(26,143,41,125)(27,144,42,126)(28,145,36,120)(43,108,211,117)(44,109,212,118)(45,110,213,119)(46,111,214,113)(47,112,215,114)(48,106,216,115)(49,107,217,116)(50,171,59,162)(51,172,60,163)(52,173,61,164)(53,174,62,165)(54,175,63,166)(55,169,57,167)(56,170,58,168)(64,201,89,176)(65,202,90,177)(66,203,91,178)(67,197,85,179)(68,198,86,180)(69,199,87,181)(70,200,88,182)(78,159,94,190)(79,160,95,191)(80,161,96,192)(81,155,97,193)(82,156,98,194)(83,157,92,195)(84,158,93,196), (1,115,59,151)(2,116,60,152)(3,117,61,153)(4,118,62,154)(5,119,63,148)(6,113,57,149)(7,114,58,150)(8,175,28,204)(9,169,22,205)(10,170,23,206)(11,171,24,207)(12,172,25,208)(13,173,26,209)(14,174,27,210)(15,198,217,161)(16,199,211,155)(17,200,212,156)(18,201,213,157)(19,202,214,158)(20,203,215,159)(21,197,216,160)(29,179,48,191)(30,180,49,192)(31,181,43,193)(32,182,44,194)(33,176,45,195)(34,177,46,196)(35,178,47,190)(36,186,222,166)(37,187,223,167)(38,188,224,168)(39,189,218,162)(40,183,219,163)(41,184,220,164)(42,185,221,165)(50,133,77,106)(51,127,71,107)(52,128,72,108)(53,129,73,109)(54,130,74,110)(55,131,75,111)(56,132,76,112)(64,139,83,120)(65,140,84,121)(66,134,78,122)(67,135,79,123)(68,136,80,124)(69,137,81,125)(70,138,82,126)(85,104,95,141)(86,105,96,142)(87,99,97,143)(88,100,98,144)(89,101,92,145)(90,102,93,146)(91,103,94,147) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49),(50,51,52,53,54,55,56),(57,58,59,60,61,62,63),(64,65,66,67,68,69,70),(71,72,73,74,75,76,77),(78,79,80,81,82,83,84),(85,86,87,88,89,90,91),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112),(113,114,115,116,117,118,119),(120,121,122,123,124,125,126),(127,128,129,130,131,132,133),(134,135,136,137,138,139,140),(141,142,143,144,145,146,147),(148,149,150,151,152,153,154),(155,156,157,158,159,160,161),(162,163,164,165,166,167,168),(169,170,171,172,173,174,175),(176,177,178,179,180,181,182),(183,184,185,186,187,188,189),(190,191,192,193,194,195,196),(197,198,199,200,201,202,203),(204,205,206,207,208,209,210),(211,212,213,214,215,216,217),(218,219,220,221,222,223,224)], [(1,77),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,33),(9,34),(10,35),(11,29),(12,30),(13,31),(14,32),(15,219),(16,220),(17,221),(18,222),(19,223),(20,224),(21,218),(22,46),(23,47),(24,48),(25,49),(26,43),(27,44),(28,45),(36,213),(37,214),(38,215),(39,216),(40,217),(41,211),(42,212),(50,59),(51,60),(52,61),(53,62),(54,63),(55,57),(56,58),(64,89),(65,90),(66,91),(67,85),(68,86),(69,87),(70,88),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(99,108),(100,109),(101,110),(102,111),(103,112),(104,106),(105,107),(113,140),(114,134),(115,135),(116,136),(117,137),(118,138),(119,139),(120,148),(121,149),(122,150),(123,151),(124,152),(125,153),(126,154),(127,142),(128,143),(129,144),(130,145),(131,146),(132,147),(133,141),(155,181),(156,182),(157,176),(158,177),(159,178),(160,179),(161,180),(162,207),(163,208),(164,209),(165,210),(166,204),(167,205),(168,206),(169,187),(170,188),(171,189),(172,183),(173,184),(174,185),(175,186),(190,203),(191,197),(192,198),(193,199),(194,200),(195,201),(196,202)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,57),(7,58),(8,28),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(15,217),(16,211),(17,212),(18,213),(19,214),(20,215),(21,216),(29,48),(30,49),(31,43),(32,44),(33,45),(34,46),(35,47),(36,222),(37,223),(38,224),(39,218),(40,219),(41,220),(42,221),(50,77),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(64,83),(65,84),(66,78),(67,79),(68,80),(69,81),(70,82),(85,95),(86,96),(87,97),(88,98),(89,92),(90,93),(91,94),(99,143),(100,144),(101,145),(102,146),(103,147),(104,141),(105,142),(106,133),(107,127),(108,128),(109,129),(110,130),(111,131),(112,132),(113,149),(114,150),(115,151),(116,152),(117,153),(118,154),(119,148),(120,139),(121,140),(122,134),(123,135),(124,136),(125,137),(126,138),(155,199),(156,200),(157,201),(158,202),(159,203),(160,197),(161,198),(162,189),(163,183),(164,184),(165,185),(166,186),(167,187),(168,188),(169,205),(170,206),(171,207),(172,208),(173,209),(174,210),(175,204),(176,195),(177,196),(178,190),(179,191),(180,192),(181,193),(182,194)], [(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,213),(9,214),(10,215),(11,216),(12,217),(13,211),(14,212),(15,25),(16,26),(17,27),(18,28),(19,22),(20,23),(21,24),(29,39),(30,40),(31,41),(32,42),(33,36),(34,37),(35,38),(43,220),(44,221),(45,222),(46,223),(47,224),(48,218),(49,219),(50,79),(51,80),(52,81),(53,82),(54,83),(55,84),(56,78),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,92),(64,74),(65,75),(66,76),(67,77),(68,71),(69,72),(70,73),(99,117),(100,118),(101,119),(102,113),(103,114),(104,115),(105,116),(106,135),(107,136),(108,137),(109,138),(110,139),(111,140),(112,134),(120,130),(121,131),(122,132),(123,133),(124,127),(125,128),(126,129),(141,151),(142,152),(143,153),(144,154),(145,148),(146,149),(147,150),(155,173),(156,174),(157,175),(158,169),(159,170),(160,171),(161,172),(162,191),(163,192),(164,193),(165,194),(166,195),(167,196),(168,190),(176,186),(177,187),(178,188),(179,189),(180,183),(181,184),(182,185),(197,207),(198,208),(199,209),(200,210),(201,204),(202,205),(203,206)], [(1,77),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,222),(9,223),(10,224),(11,218),(12,219),(13,220),(14,221),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(21,29),(22,37),(23,38),(24,39),(25,40),(26,41),(27,42),(28,36),(43,211),(44,212),(45,213),(46,214),(47,215),(48,216),(49,217),(50,59),(51,60),(52,61),(53,62),(54,63),(55,57),(56,58),(64,89),(65,90),(66,91),(67,85),(68,86),(69,87),(70,88),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(99,137),(100,138),(101,139),(102,140),(103,134),(104,135),(105,136),(106,115),(107,116),(108,117),(109,118),(110,119),(111,113),(112,114),(120,145),(121,146),(122,147),(123,141),(124,142),(125,143),(126,144),(127,152),(128,153),(129,154),(130,148),(131,149),(132,150),(133,151),(155,193),(156,194),(157,195),(158,196),(159,190),(160,191),(161,192),(162,171),(163,172),(164,173),(165,174),(166,175),(167,169),(168,170),(176,201),(177,202),(178,203),(179,197),(180,198),(181,199),(182,200),(183,208),(184,209),(185,210),(186,204),(187,205),(188,206),(189,207)], [(1,189,77,207),(2,183,71,208),(3,184,72,209),(4,185,73,210),(5,186,74,204),(6,187,75,205),(7,188,76,206),(8,101,222,139),(9,102,223,140),(10,103,224,134),(11,104,218,135),(12,105,219,136),(13,99,220,137),(14,100,221,138),(15,152,30,127),(16,153,31,128),(17,154,32,129),(18,148,33,130),(19,149,34,131),(20,150,35,132),(21,151,29,133),(22,146,37,121),(23,147,38,122),(24,141,39,123),(25,142,40,124),(26,143,41,125),(27,144,42,126),(28,145,36,120),(43,108,211,117),(44,109,212,118),(45,110,213,119),(46,111,214,113),(47,112,215,114),(48,106,216,115),(49,107,217,116),(50,171,59,162),(51,172,60,163),(52,173,61,164),(53,174,62,165),(54,175,63,166),(55,169,57,167),(56,170,58,168),(64,201,89,176),(65,202,90,177),(66,203,91,178),(67,197,85,179),(68,198,86,180),(69,199,87,181),(70,200,88,182),(78,159,94,190),(79,160,95,191),(80,161,96,192),(81,155,97,193),(82,156,98,194),(83,157,92,195),(84,158,93,196)], [(1,115,59,151),(2,116,60,152),(3,117,61,153),(4,118,62,154),(5,119,63,148),(6,113,57,149),(7,114,58,150),(8,175,28,204),(9,169,22,205),(10,170,23,206),(11,171,24,207),(12,172,25,208),(13,173,26,209),(14,174,27,210),(15,198,217,161),(16,199,211,155),(17,200,212,156),(18,201,213,157),(19,202,214,158),(20,203,215,159),(21,197,216,160),(29,179,48,191),(30,180,49,192),(31,181,43,193),(32,182,44,194),(33,176,45,195),(34,177,46,196),(35,178,47,190),(36,186,222,166),(37,187,223,167),(38,188,224,168),(39,189,218,162),(40,183,219,163),(41,184,220,164),(42,185,221,165),(50,133,77,106),(51,127,71,107),(52,128,72,108),(53,129,73,109),(54,130,74,110),(55,131,75,111),(56,132,76,112),(64,139,83,120),(65,140,84,121),(66,134,78,122),(67,135,79,123),(68,136,80,124),(69,137,81,125),(70,138,82,126),(85,104,95,141),(86,105,96,142),(87,99,97,143),(88,100,98,144),(89,101,92,145),(90,102,93,146),(91,103,94,147)]])
 

196 conjugacy classes

class 1 2A···2G2H2I4A···4L4M···4R7A···7F14A···14AP14AQ···14BB28A···28BT28BU···28DD
order12···2224···44···47···714···1414···1428···2828···28
size11···1442···24···41···11···14···42···24···4

196 irreducible representations

dim1111111111112222
type++++++
imageC1C2C2C2C2C4C7C14C14C14C14C28D4C4○D4C7×D4C7×C4○D4
kernelC7×C24.C22C7×C2.C42C2×C4×C28C14×C22⋊C4C14×C4⋊C4C7×C22⋊C4C24.C22C2.C42C2×C42C2×C22⋊C4C2×C4⋊C4C22⋊C4C2×C28C2×C14C2×C4C22
# reps121318612618648482448

Matrix representation of C7×C24.C22 ►in GL5(𝔽29)

10000
025000
002500
00010
00001
,
10000
028000
00100
00010
0001028
,
10000
028000
002800
00010
00001
,
10000
028000
002800
000280
000028
,
280000
028000
002800
00010
00001
,
170000
001700
017000
000280
000191
,
10000
00100
028000
0001027
000619

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,25,0,0,0,0,0,25,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1,10,0,0,0,0,28],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[17,0,0,0,0,0,0,17,0,0,0,17,0,0,0,0,0,0,28,19,0,0,0,0,1],[1,0,0,0,0,0,0,28,0,0,0,1,0,0,0,0,0,0,10,6,0,0,0,27,19] >;
 

C7×C24.C22 in GAP, Magma, Sage, TeX 

C_7\times C_2^4.C_2^2
 
% in TeX
 
G:=Group("C7xC2^4.C2^2");
 
// GroupNames label
 
G:=SmallGroup(448,796);
 
// by ID
 
G=gap.SmallGroup(448,796);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,1968,2438,310]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=e^2=1,f^2=e,g^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,f*b*f^-1=b*c=c*b,g*b*g^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g^-1=d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e>;
 
// generators/relations
 

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