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G = C7×C23.34D4  order 448 = 26·7

Direct product of C7 and C23.34D4

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

Aliases: C7×C23.34D4, (C22×C4)⋊7C28, (C22×C28)⋊13C4, (C23×C4).5C14, (C23×C28).6C2, C23.34(C7×D4), C24.28(C2×C14), C23.26(C2×C28), C22.31(D4×C14), C2.C42⋊2C14, (C22×C14).154D4, (C23×C14).85C22, C22.30(C22×C28), C23.54(C22×C14), C14.54(C42⋊C2), (C22×C14).445C23, (C22×C28).490C22, C14.85(C22.D4), (C2×C4).56(C2×C28), C2.6(C14×C22⋊C4), (C2×C28).284(C2×C4), (C2×C14).598(C2×D4), (C2×C22⋊C4).4C14, C14.93(C2×C22⋊C4), (C22×C4).3(C2×C14), C2.6(C7×C42⋊C2), C22.16(C7×C4○D4), (C14×C22⋊C4).10C2, (C7×C2.C42)⋊4C2, C22.16(C7×C22⋊C4), (C2×C14).206(C4○D4), C2.1(C7×C22.D4), (C2×C14).79(C22⋊C4), (C2×C14).217(C22×C4), (C22×C14).112(C2×C4), SmallGroup(448,789)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C23.34D4
Chief series
C1 — C2 — C22 — C23 — C22×C14 — C22×C28 — C14×C22⋊C4 — C7×C23.34D4
Lower central
C1 — C22 — C7×C23.34D4
Upper central
C1 — C22×C14 — C7×C23.34D4

Generators and relations for C7×C23.34D4
 G = < a,b,c,d,e,f | a7=b2=c2=d2=e4=1, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf-1=bc=cb, bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=de-1 >

Subgroups: 354 in 218 conjugacy classes, 98 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C22⋊C4, C22×C4, C22×C4, C24, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C23×C4, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C23.34D4, C7×C22⋊C4, C22×C28, C22×C28, C23×C14, C7×C2.C42, C14×C22⋊C4, C23×C28, C7×C23.34D4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, C22×C4, C2×D4, C4○D4, C28, C2×C14, C2×C22⋊C4, C42⋊C2, C22.D4, C2×C28, C7×D4, C22×C14, C23.34D4, C7×C22⋊C4, C22×C28, D4×C14, C7×C4○D4, C14×C22⋊C4, C7×C42⋊C2, C7×C22.D4, C7×C23.34D4

Smallest permutation representation of C7×C23.34D4
►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 76)(2 77)(3 71)(4 72)(5 73)(6 74)(7 75)(8 26)(9 27)(10 28)(11 22)(12 23)(13 24)(14 25)(15 220)(16 221)(17 222)(18 223)(19 224)(20 218)(21 219)(29 46)(30 47)(31 48)(32 49)(33 43)(34 44)(35 45)(36 212)(37 213)(38 214)(39 215)(40 216)(41 217)(42 211)(50 95)(51 96)(52 97)(53 98)(54 92)(55 93)(56 94)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 82)(65 83)(66 84)(67 78)(68 79)(69 80)(70 81)(99 117)(100 118)(101 119)(102 113)(103 114)(104 115)(105 116)(106 151)(107 152)(108 153)(109 154)(110 148)(111 149)(112 150)(120 138)(121 139)(122 140)(123 134)(124 135)(125 136)(126 137)(127 145)(128 146)(129 147)(130 141)(131 142)(132 143)(133 144)(155 201)(156 202)(157 203)(158 197)(159 198)(160 199)(161 200)(162 179)(163 180)(164 181)(165 182)(166 176)(167 177)(168 178)(169 186)(170 187)(171 188)(172 189)(173 183)(174 184)(175 185)(190 207)(191 208)(192 209)(193 210)(194 204)(195 205)(196 206)

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

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

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

196 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P7A···7F14A···14AP14AQ···14BN28A···28AV28AW···28CR
order12···222224···44···47···714···1414···1428···2828···28
size11···122222···24···41···11···12···22···24···4

196 irreducible representations

dim11111111112222
type+++++
imageC1C2C2C2C4C7C14C14C14C28D4C4○D4C7×D4C7×C4○D4
kernelC7×C23.34D4C7×C2.C42C14×C22⋊C4C23×C28C22×C28C23.34D4C2.C42C2×C22⋊C4C23×C4C22×C4C22×C14C2×C14C23C22
# reps1421862412648482448

Matrix representation of C7×C23.34D4 ►in GL5(𝔽29)

10000
023000
002300
000240
000024
,
280000
028000
00100
00010
0001628
,
10000
028000
002800
000280
000028
,
280000
01000
00100
000280
000028
,
170000
001400
02000
000211
000228
,
120000
002300
05000
000917
0002620

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,23,0,0,0,0,0,23,0,0,0,0,0,24,0,0,0,0,0,24],[28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1,16,0,0,0,0,28],[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],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[17,0,0,0,0,0,0,2,0,0,0,14,0,0,0,0,0,0,21,22,0,0,0,1,8],[12,0,0,0,0,0,0,5,0,0,0,23,0,0,0,0,0,0,9,26,0,0,0,17,20] >;
 

C7×C23.34D4 in GAP, Magma, Sage, TeX 

C_7\times C_2^3._{34}D_4
 
% in TeX
 
G:=Group("C7xC2^3.34D4");
 
// GroupNames label
 
G:=SmallGroup(448,789);
 
// by ID
 
G=gap.SmallGroup(448,789);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,2438,310]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^4=1,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=f*b*f^-1=b*c=c*b,b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^-1>;
 
// generators/relations
 

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