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## G = C2×C4×Dic6order 192 = 26·3

### Direct product of C2×C4 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C4×Dic6
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C22×Dic6 — C2×C4×Dic6
 Lower central C3 — C6 — C2×C4×Dic6
 Upper central C1 — C22×C4 — C2×C42

Generators and relations for C2×C4×Dic6
G = < a,b,c,d | a2=b4=c12=1, d2=c6, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 536 in 298 conjugacy classes, 183 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2×C42, C2×C42, C2×C4⋊C4, C4×Q8, C22×Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×Dic6, C22×Dic3, C22×C12, C2×C4×Q8, C4×Dic6, C2×C4×Dic3, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×C4×C12, C22×Dic6, C2×C4×Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, C22×C4, C2×Q8, C4○D4, C24, Dic6, C4×S3, C22×S3, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, C2×Dic6, S3×C2×C4, C4○D12, S3×C23, C2×C4×Q8, C4×Dic6, C22×Dic6, S3×C22×C4, C2×C4○D12, C2×C4×Dic6

Smallest permutation representation of C2×C4×Dic6
Regular action on 192 points
Generators in S192
(1 170)(2 171)(3 172)(4 173)(5 174)(6 175)(7 176)(8 177)(9 178)(10 179)(11 180)(12 169)(13 185)(14 186)(15 187)(16 188)(17 189)(18 190)(19 191)(20 192)(21 181)(22 182)(23 183)(24 184)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 73)(34 74)(35 75)(36 76)(37 132)(38 121)(39 122)(40 123)(41 124)(42 125)(43 126)(44 127)(45 128)(46 129)(47 130)(48 131)(49 115)(50 116)(51 117)(52 118)(53 119)(54 120)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 85)(69 86)(70 87)(71 88)(72 89)(97 166)(98 167)(99 168)(100 157)(101 158)(102 159)(103 160)(104 161)(105 162)(106 163)(107 164)(108 165)(133 154)(134 155)(135 156)(136 145)(137 146)(138 147)(139 148)(140 149)(141 150)(142 151)(143 152)(144 153)
(1 23 138 46)(2 24 139 47)(3 13 140 48)(4 14 141 37)(5 15 142 38)(6 16 143 39)(7 17 144 40)(8 18 133 41)(9 19 134 42)(10 20 135 43)(11 21 136 44)(12 22 137 45)(25 64 58 101)(26 65 59 102)(27 66 60 103)(28 67 49 104)(29 68 50 105)(30 69 51 106)(31 70 52 107)(32 71 53 108)(33 72 54 97)(34 61 55 98)(35 62 56 99)(36 63 57 100)(73 89 120 166)(74 90 109 167)(75 91 110 168)(76 92 111 157)(77 93 112 158)(78 94 113 159)(79 95 114 160)(80 96 115 161)(81 85 116 162)(82 86 117 163)(83 87 118 164)(84 88 119 165)(121 174 187 151)(122 175 188 152)(123 176 189 153)(124 177 190 154)(125 178 191 155)(126 179 192 156)(127 180 181 145)(128 169 182 146)(129 170 183 147)(130 171 184 148)(131 172 185 149)(132 173 186 150)
(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 91 7 85)(2 90 8 96)(3 89 9 95)(4 88 10 94)(5 87 11 93)(6 86 12 92)(13 120 19 114)(14 119 20 113)(15 118 21 112)(16 117 22 111)(17 116 23 110)(18 115 24 109)(25 121 31 127)(26 132 32 126)(27 131 33 125)(28 130 34 124)(29 129 35 123)(30 128 36 122)(37 84 43 78)(38 83 44 77)(39 82 45 76)(40 81 46 75)(41 80 47 74)(42 79 48 73)(49 184 55 190)(50 183 56 189)(51 182 57 188)(52 181 58 187)(53 192 59 186)(54 191 60 185)(61 177 67 171)(62 176 68 170)(63 175 69 169)(64 174 70 180)(65 173 71 179)(66 172 72 178)(97 155 103 149)(98 154 104 148)(99 153 105 147)(100 152 106 146)(101 151 107 145)(102 150 108 156)(133 161 139 167)(134 160 140 166)(135 159 141 165)(136 158 142 164)(137 157 143 163)(138 168 144 162)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 4I ··· 4P 4Q ··· 4AF 6A ··· 6G 12A ··· 12X order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C4 S3 Q8 D6 D6 C4○D4 Dic6 C4×S3 C4○D12 kernel C2×C4×Dic6 C4×Dic6 C2×C4×Dic3 C2×Dic3⋊C4 C2×C4⋊Dic3 C2×C4×C12 C22×Dic6 C2×Dic6 C2×C42 C2×C12 C42 C22×C4 C2×C6 C2×C4 C2×C4 C22 # reps 1 8 2 2 1 1 1 16 1 4 4 3 4 8 8 8

Matrix representation of C2×C4×Dic6 in GL6(𝔽13)

 1 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 12 0 0 0 0 0 0 12
,
 8 0 0 0 0 0 0 8 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
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 0 0 0 9 0 0 0 0 0 0 3
,
 4 3 0 0 0 0 3 9 0 0 0 0 0 0 3 9 0 0 0 0 9 10 0 0 0 0 0 0 0 12 0 0 0 0 12 0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[4,3,0,0,0,0,3,9,0,0,0,0,0,0,3,9,0,0,0,0,9,10,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;

C2×C4×Dic6 in GAP, Magma, Sage, TeX

C_2\times C_4\times {\rm Dic}_6
% in TeX

G:=Group("C2xC4xDic6");
// GroupNames label

G:=SmallGroup(192,1026);
// by ID

G=gap.SmallGroup(192,1026);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,184,80,6278]);
// Polycyclic

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

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