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## G = C2×C2.Dic12order 192 = 26·3

### Direct product of C2 and C2.Dic12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×C2.Dic12
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C2×C4⋊Dic3 — C2×C2.Dic12
 Lower central C3 — C6 — C12 — C2×C2.Dic12
 Upper central C1 — C23 — C22×C4 — C22×C8

Generators and relations for C2×C2.Dic12
G = < a,b,c,d | a2=b6=c8=1, d2=b3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b3c3 >

Subgroups: 408 in 162 conjugacy classes, 79 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, C24, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, Q8⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C4⋊Dic3, C4⋊Dic3, C2×C24, C2×C24, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C2×Q8⋊C4, C2.Dic12, C2×C4⋊Dic3, C22×C24, C22×Dic6, C2×C2.Dic12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, SD16, Q16, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C24⋊C2, Dic12, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C2×Q8⋊C4, C2.Dic12, C2×C24⋊C2, C2×Dic12, C2×D6⋊C4, C2×C2.Dic12

Smallest permutation representation of C2×C2.Dic12
Regular action on 192 points
Generators in S192
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 108)(18 109)(19 110)(20 111)(21 112)(22 105)(23 106)(24 107)(25 140)(26 141)(27 142)(28 143)(29 144)(30 137)(31 138)(32 139)(33 70)(34 71)(35 72)(36 65)(37 66)(38 67)(39 68)(40 69)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 121)(49 154)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 153)(57 84)(58 85)(59 86)(60 87)(61 88)(62 81)(63 82)(64 83)(73 173)(74 174)(75 175)(76 176)(77 169)(78 170)(79 171)(80 172)(89 192)(90 185)(91 186)(92 187)(93 188)(94 189)(95 190)(96 191)(97 149)(98 150)(99 151)(100 152)(101 145)(102 146)(103 147)(104 148)(113 164)(114 165)(115 166)(116 167)(117 168)(118 161)(119 162)(120 163)(129 181)(130 182)(131 183)(132 184)(133 177)(134 178)(135 179)(136 180)
(1 59 35 159 137 125)(2 60 36 160 138 126)(3 61 37 153 139 127)(4 62 38 154 140 128)(5 63 39 155 141 121)(6 64 40 156 142 122)(7 57 33 157 143 123)(8 58 34 158 144 124)(9 81 67 49 25 47)(10 82 68 50 26 48)(11 83 69 51 27 41)(12 84 70 52 28 42)(13 85 71 53 29 43)(14 86 72 54 30 44)(15 87 65 55 31 45)(16 88 66 56 32 46)(17 145 168 135 77 89)(18 146 161 136 78 90)(19 147 162 129 79 91)(20 148 163 130 80 92)(21 149 164 131 73 93)(22 150 165 132 74 94)(23 151 166 133 75 95)(24 152 167 134 76 96)(97 113 183 173 188 112)(98 114 184 174 189 105)(99 115 177 175 190 106)(100 116 178 176 191 107)(101 117 179 169 192 108)(102 118 180 170 185 109)(103 119 181 171 186 110)(104 120 182 172 187 111)
(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)
(1 129 159 19)(2 22 160 132)(3 135 153 17)(4 20 154 130)(5 133 155 23)(6 18 156 136)(7 131 157 21)(8 24 158 134)(9 111 49 182)(10 177 50 106)(11 109 51 180)(12 183 52 112)(13 107 53 178)(14 181 54 110)(15 105 55 184)(16 179 56 108)(25 120 81 187)(26 190 82 115)(27 118 83 185)(28 188 84 113)(29 116 85 191)(30 186 86 119)(31 114 87 189)(32 192 88 117)(33 149 123 73)(34 76 124 152)(35 147 125 79)(36 74 126 150)(37 145 127 77)(38 80 128 148)(39 151 121 75)(40 78 122 146)(41 102 69 170)(42 173 70 97)(43 100 71 176)(44 171 72 103)(45 98 65 174)(46 169 66 101)(47 104 67 172)(48 175 68 99)(57 164 143 93)(58 96 144 167)(59 162 137 91)(60 94 138 165)(61 168 139 89)(62 92 140 163)(63 166 141 95)(64 90 142 161)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E ··· 4L 6A ··· 6G 8A ··· 8H 12A ··· 12H 24A ··· 24P order 1 2 ··· 2 3 4 4 4 4 4 ··· 4 6 ··· 6 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 ··· 1 2 2 2 2 2 12 ··· 12 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C4 S3 D4 D4 D6 D6 SD16 Q16 C4×S3 D12 C3⋊D4 D12 C24⋊C2 Dic12 kernel C2×C2.Dic12 C2.Dic12 C2×C4⋊Dic3 C22×C24 C22×Dic6 C2×Dic6 C22×C8 C2×C12 C22×C6 C2×C8 C22×C4 C2×C6 C2×C6 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 1 3 1 2 1 4 4 4 2 4 2 8 8

Matrix representation of C2×C2.Dic12 in GL4(𝔽73) generated by

 1 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 0 1 0 0 0 0 0 72 0 0 1 1
,
 27 0 0 0 0 1 0 0 0 0 18 68 0 0 5 23
,
 46 0 0 0 0 72 0 0 0 0 62 37 0 0 48 11
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,1,0,0,0,0,0,1,0,0,72,1],[27,0,0,0,0,1,0,0,0,0,18,5,0,0,68,23],[46,0,0,0,0,72,0,0,0,0,62,48,0,0,37,11] >;

C2×C2.Dic12 in GAP, Magma, Sage, TeX

C_2\times C_2.{\rm Dic}_{12}
% in TeX

G:=Group("C2xC2.Dic12");
// GroupNames label

G:=SmallGroup(192,662);
// by ID

G=gap.SmallGroup(192,662);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,254,142,1123,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^8=1,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^3>;
// generators/relations

׿
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