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G = C2×C42⋊2D7  order 448 = 26·7

Direct product of C2 and C42⋊2D7

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C42⋊2D7, C42⋊36D14, (C2×C42)⋊4D7, (C4×C28)⋊47C22, (C2×C14).22C24, C14⋊1(C42⋊2C2), (C2×C28).695C23, Dic7⋊C4⋊41C22, D14⋊C4.81C22, (C22×C4).409D14, (C2×Dic7).6C23, (C22×D7).4C23, C22.65(C23×D7), C22.70(C4○D28), (C23×D7).29C22, C23.317(C22×D7), (C22×C28).505C22, (C22×C14).384C23, (C22×Dic7).75C22, (C2×C4×C28)⋊4C2, C7⋊1(C2×C42⋊2C2), C14.9(C2×C4○D4), C2.11(C2×C4○D28), (C2×Dic7⋊C4)⋊16C2, (C2×D14⋊C4).16C2, (C2×C14).98(C4○D4), (C2×C4).650(C22×D7), SmallGroup(448,931)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C2×C42⋊2D7
Chief series
C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C2×D14⋊C4 — C2×C42⋊2D7
Lower central
C7 — C2×C14 — C2×C42⋊2D7
Upper central
C1 — C23 — C2×C42

Generators and relations for C2×C42⋊2D7
 G = < a,b,c,d,e | a2=b4=c4=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=bc2, cd=dc, ece=b2c-1, ede=d-1 >

Subgroups: 1092 in 246 conjugacy classes, 111 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊2C2, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C2×C42⋊2C2, Dic7⋊C4, D14⋊C4, C4×C28, C22×Dic7, C22×C28, C23×D7, C42⋊2D7, C2×Dic7⋊C4, C2×D14⋊C4, C2×C4×C28, C2×C42⋊2D7
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C42⋊2C2, C2×C4○D4, C22×D7, C2×C42⋊2C2, C4○D28, C23×D7, C42⋊2D7, C2×C4○D28, C2×C42⋊2D7

Smallest permutation representation of C2×C42⋊2D7
►On 224 points
Generators in S224
(1 120)(2 121)(3 122)(4 123)(5 124)(6 125)(7 126)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 134)(16 135)(17 136)(18 137)(19 138)(20 139)(21 140)(22 127)(23 128)(24 129)(25 130)(26 131)(27 132)(28 133)(29 148)(30 149)(31 150)(32 151)(33 152)(34 153)(35 154)(36 141)(37 142)(38 143)(39 144)(40 145)(41 146)(42 147)(43 162)(44 163)(45 164)(46 165)(47 166)(48 167)(49 168)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 161)(57 176)(58 177)(59 178)(60 179)(61 180)(62 181)(63 182)(64 169)(65 170)(66 171)(67 172)(68 173)(69 174)(70 175)(71 190)(72 191)(73 192)(74 193)(75 194)(76 195)(77 196)(78 183)(79 184)(80 185)(81 186)(82 187)(83 188)(84 189)(85 204)(86 205)(87 206)(88 207)(89 208)(90 209)(91 210)(92 197)(93 198)(94 199)(95 200)(96 201)(97 202)(98 203)(99 218)(100 219)(101 220)(102 221)(103 222)(104 223)(105 224)(106 211)(107 212)(108 213)(109 214)(110 215)(111 216)(112 217)

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

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

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

124 conjugacy classes

class 1 2A···2G2H2I4A···4L4M···4R7A7B7C14A···14U28A···28BT
order12···2224···44···477714···1428···28
size11···128282···228···282222···22···2

124 irreducible representations

dim1111122222
type++++++++
imageC1C2C2C2C2D7C4○D4D14D14C4○D28
kernelC2×C42⋊2D7C42⋊2D7C2×Dic7⋊C4C2×D14⋊C4C2×C4×C28C2×C42C2×C14C42C22×C4C22
# reps1833131212972

Matrix representation of C2×C42⋊2D7 ►in GL5(𝔽29)

280000
01000
00100
000280
000028
,
280000
001200
012000
000120
000012
,
280000
017000
001700
0001816
000711
,
10000
01000
00100
000428
000528
,
280000
01000
002800
0001126
0001118

G:=sub<GL(5,GF(29))| [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],[28,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12],[28,0,0,0,0,0,17,0,0,0,0,0,17,0,0,0,0,0,18,7,0,0,0,16,11],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,5,0,0,0,28,28],[28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,11,11,0,0,0,26,18] >;
 

C2×C42⋊2D7 in GAP, Magma, Sage, TeX 

C_2\times C_4^2\rtimes_2D_7
 
% in TeX
 
G:=Group("C2xC4^2:2D7");
 
// GroupNames label
 
G:=SmallGroup(448,931);
 
// by ID
 
G=gap.SmallGroup(448,931);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,100,1571,136,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=b*c^2,c*d=d*c,e*c*e=b^2*c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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