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G = C2×D26⋊C4order 416 = 25·13

Direct product of C2 and D26⋊C4

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

Aliases: C2×D26⋊C4, C23.31D26, C22.16D52, (C2×C4)⋊8D26, D267(C2×C4), C2.3(C2×D52), (C22×C52)⋊1C2, (C2×C26).36D4, C26.40(C2×D4), (C22×C4)⋊1D13, C262(C22⋊C4), (C2×C52)⋊10C22, (C22×D13)⋊4C4, (C2×C26).45C23, C26.31(C22×C4), (C23×D13).2C2, C22.17(C4×D13), (C22×Dic13)⋊3C2, (C2×Dic13)⋊6C22, C22.20(C13⋊D4), (C22×C26).37C22, C22.23(C22×D13), (C22×D13).26C22, C133(C2×C22⋊C4), C2.19(C2×C4×D13), C2.2(C2×C13⋊D4), (C2×C26).38(C2×C4), SmallGroup(416,148)

Series: Derived Chief Lower central Upper central

C1C26 — C2×D26⋊C4
C1C13C26C2×C26C22×D13C23×D13 — C2×D26⋊C4
C13C26 — C2×D26⋊C4
C1C23C22×C4

Generators and relations for C2×D26⋊C4
 G = < a,b,c,d | a2=b26=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b13c >

Subgroups: 992 in 132 conjugacy classes, 57 normal (17 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C22, C22 [×6], C22 [×16], C2×C4 [×2], C2×C4 [×6], C23, C23 [×10], C13, C22⋊C4 [×4], C22×C4, C22×C4, C24, D13 [×4], C26 [×3], C26 [×4], C2×C22⋊C4, Dic13 [×2], C52 [×2], D26 [×4], D26 [×12], C2×C26, C2×C26 [×6], C2×Dic13 [×2], C2×Dic13 [×2], C2×C52 [×2], C2×C52 [×2], C22×D13 [×6], C22×D13 [×4], C22×C26, D26⋊C4 [×4], C22×Dic13, C22×C52, C23×D13, C2×D26⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D13, C2×C22⋊C4, D26 [×3], C4×D13 [×2], D52 [×2], C13⋊D4 [×2], C22×D13, D26⋊C4 [×4], C2×C4×D13, C2×D52, C2×C13⋊D4, C2×D26⋊C4

Smallest permutation representation of C2×D26⋊C4
On 208 points
Generators in S208
(1 134)(2 135)(3 136)(4 137)(5 138)(6 139)(7 140)(8 141)(9 142)(10 143)(11 144)(12 145)(13 146)(14 147)(15 148)(16 149)(17 150)(18 151)(19 152)(20 153)(21 154)(22 155)(23 156)(24 131)(25 132)(26 133)(27 116)(28 117)(29 118)(30 119)(31 120)(32 121)(33 122)(34 123)(35 124)(36 125)(37 126)(38 127)(39 128)(40 129)(41 130)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 188)(54 189)(55 190)(56 191)(57 192)(58 193)(59 194)(60 195)(61 196)(62 197)(63 198)(64 199)(65 200)(66 201)(67 202)(68 203)(69 204)(70 205)(71 206)(72 207)(73 208)(74 183)(75 184)(76 185)(77 186)(78 187)(79 176)(80 177)(81 178)(82 179)(83 180)(84 181)(85 182)(86 157)(87 158)(88 159)(89 160)(90 161)(91 162)(92 163)(93 164)(94 165)(95 166)(96 167)(97 168)(98 169)(99 170)(100 171)(101 172)(102 173)(103 174)(104 175)
(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)
(1 126)(2 125)(3 124)(4 123)(5 122)(6 121)(7 120)(8 119)(9 118)(10 117)(11 116)(12 115)(13 114)(14 113)(15 112)(16 111)(17 110)(18 109)(19 108)(20 107)(21 106)(22 105)(23 130)(24 129)(25 128)(26 127)(27 144)(28 143)(29 142)(30 141)(31 140)(32 139)(33 138)(34 137)(35 136)(36 135)(37 134)(38 133)(39 132)(40 131)(41 156)(42 155)(43 154)(44 153)(45 152)(46 151)(47 150)(48 149)(49 148)(50 147)(51 146)(52 145)(53 162)(54 161)(55 160)(56 159)(57 158)(58 157)(59 182)(60 181)(61 180)(62 179)(63 178)(64 177)(65 176)(66 175)(67 174)(68 173)(69 172)(70 171)(71 170)(72 169)(73 168)(74 167)(75 166)(76 165)(77 164)(78 163)(79 200)(80 199)(81 198)(82 197)(83 196)(84 195)(85 194)(86 193)(87 192)(88 191)(89 190)(90 189)(91 188)(92 187)(93 186)(94 185)(95 184)(96 183)(97 208)(98 207)(99 206)(100 205)(101 204)(102 203)(103 202)(104 201)
(1 93 51 78)(2 94 52 53)(3 95 27 54)(4 96 28 55)(5 97 29 56)(6 98 30 57)(7 99 31 58)(8 100 32 59)(9 101 33 60)(10 102 34 61)(11 103 35 62)(12 104 36 63)(13 79 37 64)(14 80 38 65)(15 81 39 66)(16 82 40 67)(17 83 41 68)(18 84 42 69)(19 85 43 70)(20 86 44 71)(21 87 45 72)(22 88 46 73)(23 89 47 74)(24 90 48 75)(25 91 49 76)(26 92 50 77)(105 204 151 181)(106 205 152 182)(107 206 153 157)(108 207 154 158)(109 208 155 159)(110 183 156 160)(111 184 131 161)(112 185 132 162)(113 186 133 163)(114 187 134 164)(115 188 135 165)(116 189 136 166)(117 190 137 167)(118 191 138 168)(119 192 139 169)(120 193 140 170)(121 194 141 171)(122 195 142 172)(123 196 143 173)(124 197 144 174)(125 198 145 175)(126 199 146 176)(127 200 147 177)(128 201 148 178)(129 202 149 179)(130 203 150 180)

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

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

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

116 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H13A···13F26A···26AP52A···52AV
order12···222224444444413···1326···2652···52
size11···1262626262222262626262···22···22···2

116 irreducible representations

dim1111112222222
type++++++++++
imageC1C2C2C2C2C4D4D13D26D26C4×D13D52C13⋊D4
kernelC2×D26⋊C4D26⋊C4C22×Dic13C22×C52C23×D13C22×D13C2×C26C22×C4C2×C4C23C22C22C22
# reps14111846126242424

Matrix representation of C2×D26⋊C4 in GL5(𝔽53)

520000
01000
00100
000520
000052
,
10000
0393300
0342200
0001151
0004516
,
520000
0164400
0463700
00010
0002452
,
300000
030000
003000
0002520
0002728

G:=sub<GL(5,GF(53))| [52,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,52,0,0,0,0,0,52],[1,0,0,0,0,0,39,34,0,0,0,33,22,0,0,0,0,0,11,45,0,0,0,51,16],[52,0,0,0,0,0,16,46,0,0,0,44,37,0,0,0,0,0,1,24,0,0,0,0,52],[30,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,25,27,0,0,0,20,28] >;

C2×D26⋊C4 in GAP, Magma, Sage, TeX

C_2\times D_{26}\rtimes C_4
% in TeX

G:=Group("C2xD26:C4");
// GroupNames label

G:=SmallGroup(416,148);
// by ID

G=gap.SmallGroup(416,148);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,362,50,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^26=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^13*c>;
// generators/relations

׿
×
𝔽