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## G = (C2×C12).54D4order 192 = 26·3

### 28th 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).54D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×Dic3⋊C4 — (C2×C12).54D4
 Lower central C3 — C22×C6 — (C2×C12).54D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C12).54D4
G = < a,b,c,d | a2=b12=c4=1, d2=ab6, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=ab-1, dcd-1=b6c-1 >

Subgroups: 360 in 150 conjugacy classes, 63 normal (51 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C4⋊C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2×C4⋊C4, C2×C4⋊C4, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C22×Dic3, C22×C12, C23.81C23, C6.C42, C2×Dic3⋊C4, C2×C4⋊Dic3, C6×C4⋊C4, (C2×C12).54D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C3⋊D4, C22×S3, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C2×Dic6, C4○D12, S3×D4, D42S3, S3×Q8, Q83S3, C2×C3⋊D4, C23.81C23, C12⋊Q8, Dic3.Q8, C4.Dic6, D6.D4, C12.48D4, C23.14D6, D63Q8, (C2×C12).54D4

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

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

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

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

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 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + - + - + - - + image C1 C2 C2 C2 C2 S3 D4 Q8 D4 Q8 D6 C4○D4 Dic6 C3⋊D4 C4○D12 S3×D4 D4⋊2S3 S3×Q8 Q8⋊3S3 kernel (C2×C12).54D4 C6.C42 C2×Dic3⋊C4 C2×C4⋊Dic3 C6×C4⋊C4 C2×C4⋊C4 C2×Dic3 C2×Dic3 C2×C12 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C22 C22 C22 C22 C22 # reps 1 3 2 1 1 1 2 2 2 2 3 6 4 4 4 1 1 1 1

Matrix representation of (C2×C12).54D4 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 7 2 0 0 0 0 1 6 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 0 0 1 0
,
 1 5 0 0 0 0 10 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 8 0 0 0 0 0 8
,
 1 5 0 0 0 0 10 12 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 3 3 0 0 0 0 6 10

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,1,0,0,0,0,2,6,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[1,10,0,0,0,0,5,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,8,8],[1,10,0,0,0,0,5,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,3,6,0,0,0,0,3,10] >;

(C2×C12).54D4 in GAP, Magma, Sage, TeX

(C_2\times C_{12})._{54}D_4
% in TeX

G:=Group("(C2xC12).54D4");
// GroupNames label

G:=SmallGroup(192,541);
// by ID

G=gap.SmallGroup(192,541);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,344,254,387,184,6278]);
// 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=b^-1,d*b*d^-1=a*b^-1,d*c*d^-1=b^6*c^-1>;
// generators/relations

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