Copied to
clipboard

## G = (C2×C12).55D4order 192 = 26·3

### 29th non-split extension by C2×C12 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C2×C12).55D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4⋊Dic3 — (C2×C12).55D4
 Lower central C3 — C22×C6 — (C2×C12).55D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C12).55D4
G = < a,b,c,d | a2=b12=c4=1, d2=ab6, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=b6c-1 >

Subgroups: 328 in 134 conjugacy classes, 59 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×9], C22 [×3], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×4], C2×C4 [×19], C23, Dic3 [×4], C12 [×5], C2×C6 [×3], C2×C6 [×4], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×4], C2×Dic3 [×12], C2×C12 [×4], C2×C12 [×7], C22×C6, C2.C42 [×5], C2×C4⋊C4, C2×C4⋊C4, C4⋊Dic3 [×2], C3×C4⋊C4 [×2], C22×Dic3 [×4], C22×C12 [×3], C23.83C23, C6.C42, C6.C42 [×4], C2×C4⋊Dic3, C6×C4⋊C4, (C2×C12).55D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4 [×5], Dic6 [×2], C3⋊D4 [×2], C22×S3, C22⋊Q8, C22.D4, C4.4D4, C42.C2 [×2], C422C2 [×2], C2×Dic6, C4○D12, D42S3 [×2], Q83S3 [×2], C2×C3⋊D4, C23.83C23, C4.Dic6 [×2], C4⋊C4⋊S3 [×2], C12.48D4, C23.23D6, C12.23D4, (C2×C12).55D4

Smallest permutation representation of (C2×C12).55D4
Regular action on 192 points
Generators in S192
(1 108)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 186)(14 187)(15 188)(16 189)(17 190)(18 191)(19 192)(20 181)(21 182)(22 183)(23 184)(24 185)(25 173)(26 174)(27 175)(28 176)(29 177)(30 178)(31 179)(32 180)(33 169)(34 170)(35 171)(36 172)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 73)(46 74)(47 75)(48 76)(49 126)(50 127)(51 128)(52 129)(53 130)(54 131)(55 132)(56 121)(57 122)(58 123)(59 124)(60 125)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 85)(72 86)(109 158)(110 159)(111 160)(112 161)(113 162)(114 163)(115 164)(116 165)(117 166)(118 167)(119 168)(120 157)(133 147)(134 148)(135 149)(136 150)(137 151)(138 152)(139 153)(140 154)(141 155)(142 156)(143 145)(144 146)
(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)
(1 89 173 77)(2 62 174 48)(3 87 175 75)(4 72 176 46)(5 85 177 73)(6 70 178 44)(7 95 179 83)(8 68 180 42)(9 93 169 81)(10 66 170 40)(11 91 171 79)(12 64 172 38)(13 142 126 164)(14 155 127 114)(15 140 128 162)(16 153 129 112)(17 138 130 160)(18 151 131 110)(19 136 132 158)(20 149 121 120)(21 134 122 168)(22 147 123 118)(23 144 124 166)(24 145 125 116)(25 37 108 63)(26 76 97 88)(27 47 98 61)(28 74 99 86)(29 45 100 71)(30 84 101 96)(31 43 102 69)(32 82 103 94)(33 41 104 67)(34 80 105 92)(35 39 106 65)(36 78 107 90)(49 115 186 156)(50 163 187 141)(51 113 188 154)(52 161 189 139)(53 111 190 152)(54 159 191 137)(55 109 192 150)(56 157 181 135)(57 119 182 148)(58 167 183 133)(59 117 184 146)(60 165 185 143)
(1 182 102 15)(2 181 103 14)(3 192 104 13)(4 191 105 24)(5 190 106 23)(6 189 107 22)(7 188 108 21)(8 187 97 20)(9 186 98 19)(10 185 99 18)(11 184 100 17)(12 183 101 16)(25 122 179 51)(26 121 180 50)(27 132 169 49)(28 131 170 60)(29 130 171 59)(30 129 172 58)(31 128 173 57)(32 127 174 56)(33 126 175 55)(34 125 176 54)(35 124 177 53)(36 123 178 52)(37 140 83 148)(38 139 84 147)(39 138 73 146)(40 137 74 145)(41 136 75 156)(42 135 76 155)(43 134 77 154)(44 133 78 153)(45 144 79 152)(46 143 80 151)(47 142 81 150)(48 141 82 149)(61 164 93 109)(62 163 94 120)(63 162 95 119)(64 161 96 118)(65 160 85 117)(66 159 86 116)(67 158 87 115)(68 157 88 114)(69 168 89 113)(70 167 90 112)(71 166 91 111)(72 165 92 110)

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4F 4G ··· 4N 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 4 ··· 4 12 ··· 12 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + - + - - + image C1 C2 C2 C2 S3 D4 Q8 D6 C4○D4 Dic6 C3⋊D4 C4○D12 D4⋊2S3 Q8⋊3S3 kernel (C2×C12).55D4 C6.C42 C2×C4⋊Dic3 C6×C4⋊C4 C2×C4⋊C4 C2×C12 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C22 C22 C22 # reps 1 5 1 1 1 2 2 3 10 4 4 4 2 2

Matrix representation of (C2×C12).55D4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 3 0 0 0 0 8 1 0 0 0 0 0 0 4 0 0 0 0 0 5 10 0 0 0 0 0 0 6 0 0 0 0 0 0 11
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 11 8 0 0 0 0 1 2 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 5 12 0 0 0 0 0 0 11 8 0 0 0 0 11 2 0 0 0 0 0 0 0 1 0 0 0 0 12 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,8,0,0,0,0,3,1,0,0,0,0,0,0,4,5,0,0,0,0,0,10,0,0,0,0,0,0,6,0,0,0,0,0,0,11],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,11,1,0,0,0,0,8,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,5,0,0,0,0,0,12,0,0,0,0,0,0,11,11,0,0,0,0,8,2,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;

(C2×C12).55D4 in GAP, Magma, Sage, TeX

(C_2\times C_{12})._{55}D_4
% in TeX

G:=Group("(C2xC12).55D4");
// GroupNames label

G:=SmallGroup(192,545);
// by ID

G=gap.SmallGroup(192,545);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,701,344,254,387,184,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=a*b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=b^-1,d*c*d^-1=b^6*c^-1>;
// generators/relations

׿
×
𝔽