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G = C7×C23.11D4  order 448 = 26·7

Direct product of C7 and C23.11D4

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

Aliases: C7×C23.11D4, (C2×C28).310D4, C23.10(C7×D4), C24.11(C2×C14), C22.73(D4×C14), (C22×C14).29D4, C2.C42⋊5C14, (C23×C14).8C22, C14.140(C4⋊D4), C14.69(C4.4D4), C23.80(C22×C14), (C22×C28).35C22, C14.35(C42⋊2C2), (C22×C14).461C23, C14.91(C22.D4), (C2×C4⋊C4)⋊7C14, (C14×C4⋊C4)⋊34C2, (C2×C4).17(C7×D4), C2.9(C7×C4⋊D4), C2.7(C7×C4.4D4), (C2×C14).613(C2×D4), (C2×C22⋊C4).7C14, (C22×C4).8(C2×C14), C2.5(C7×C42⋊2C2), C22.40(C7×C4○D4), (C14×C22⋊C4).29C2, (C7×C2.C42)⋊7C2, (C2×C14).221(C4○D4), C2.7(C7×C22.D4), SmallGroup(448,805)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23 — C7×C23.11D4
Chief series
C1 — C2 — C22 — C23 — C22×C14 — C22×C28 — C14×C22⋊C4 — C7×C23.11D4
Lower central
C1 — C23 — C7×C23.11D4
Upper central
C1 — C22×C14 — C7×C23.11D4

Generators and relations for C7×C23.11D4
 G = < a,b,c,d,e,f | a7=b2=c2=d2=e4=1, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 314 in 170 conjugacy classes, 70 normal (30 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C23.11D4, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C22×C28, C23×C14, C7×C2.C42, C7×C2.C42, C14×C22⋊C4, C14×C22⋊C4, C14×C4⋊C4, C7×C23.11D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C7×D4, C22×C14, C23.11D4, D4×C14, C7×C4○D4, C7×C4⋊D4, C7×C22.D4, C7×C4.4D4, C7×C42⋊2C2, C7×C23.11D4

Smallest permutation representation of C7×C23.11D4
►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)

(8 35)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 211)(16 212)(17 213)(18 214)(19 215)(20 216)(21 217)(22 48)(23 49)(24 43)(25 44)(26 45)(27 46)(28 47)(36 221)(37 222)(38 223)(39 224)(40 218)(41 219)(42 220)(99 153)(100 154)(101 148)(102 149)(103 150)(104 151)(105 152)(106 115)(107 116)(108 117)(109 118)(110 119)(111 113)(112 114)(120 129)(121 130)(122 131)(123 132)(124 133)(125 127)(126 128)(134 143)(135 144)(136 145)(137 146)(138 147)(139 141)(140 142)(155 183)(156 184)(157 185)(158 186)(159 187)(160 188)(161 189)(162 190)(163 191)(164 192)(165 193)(166 194)(167 195)(168 196)(169 197)(170 198)(171 199)(172 200)(173 201)(174 202)(175 203)(176 204)(177 205)(178 206)(179 207)(180 208)(181 209)(182 210)

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

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

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

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

154 conjugacy classes

class 1 2A···2G2H2I4A···4L7A···7F14A···14AP14AQ···14BB28A···28BT
order12···2224···47···714···1414···1428···28
size11···1444···41···11···14···44···4

154 irreducible representations

dim11111111222222
type++++++
imageC1C2C2C2C7C14C14C14D4D4C4○D4C7×D4C7×D4C7×C4○D4
kernelC7×C23.11D4C7×C2.C42C14×C22⋊C4C14×C4⋊C4C23.11D4C2.C42C2×C22⋊C4C2×C4⋊C4C2×C28C22×C14C2×C14C2×C4C23C22
# reps13316181862210121260

Matrix representation of C7×C23.11D4 ►in GL6(𝔽29)

100000
010000
0023000
0002300
0000160
0000016
,
110000
0280000
001000
0002800
000016
0000028
,
2800000
0280000
0028000
0002800
0000280
0000028
,
2800000
0280000
0028000
0002800
000010
000001
,
1200000
5170000
0001700
0012000
0000170
0000412
,
12120000
0170000
0012000
0001200
00002823
0000101

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,0,0,0,0,0,0,23,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,1,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,6,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,5,0,0,0,0,0,17,0,0,0,0,0,0,0,12,0,0,0,0,17,0,0,0,0,0,0,0,17,4,0,0,0,0,0,12],[12,0,0,0,0,0,12,17,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,28,10,0,0,0,0,23,1] >;
 

C7×C23.11D4 in GAP, Magma, Sage, TeX 

C_7\times C_2^3._{11}D_4
 
% in TeX
 
G:=Group("C7xC2^3.11D4");
 
// GroupNames label
 
G:=SmallGroup(448,805);
 
// by ID
 
G=gap.SmallGroup(448,805);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,2438,2403,310]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^4=1,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=b*c=c*b,b*d=d*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
 
// generators/relations
 

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