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## G = C2×C12⋊2Q8order 192 = 26·3

### Direct product of C2 and C12⋊2Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C12⋊2Q8
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C22×Dic6 — C2×C12⋊2Q8
 Lower central C3 — C2×C6 — C2×C12⋊2Q8
 Upper central C1 — C23 — C2×C42

Generators and relations for C2×C122Q8
G = < a,b,c,d | a2=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 632 in 290 conjugacy classes, 159 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C2×C42, C2×C4⋊C4, C4⋊Q8, C22×Q8, C4⋊Dic3, C4×C12, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C22×C12, C2×C4⋊Q8, C122Q8, C2×C4⋊Dic3, C2×C4×C12, C22×Dic6, C2×C122Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C24, Dic6, D12, C22×S3, C4⋊Q8, C22×D4, C22×Q8, C2×Dic6, C2×D12, S3×C23, C2×C4⋊Q8, C122Q8, C22×Dic6, C22×D12, C2×C122Q8

Smallest permutation representation of C2×C122Q8
Regular action on 192 points
Generators in S192
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 37)(25 127)(26 128)(27 129)(28 130)(29 131)(30 132)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(49 106)(50 107)(51 108)(52 97)(53 98)(54 99)(55 100)(56 101)(57 102)(58 103)(59 104)(60 105)(61 179)(62 180)(63 169)(64 170)(65 171)(66 172)(67 173)(68 174)(69 175)(70 176)(71 177)(72 178)(85 148)(86 149)(87 150)(88 151)(89 152)(90 153)(91 154)(92 155)(93 156)(94 145)(95 146)(96 147)(109 137)(110 138)(111 139)(112 140)(113 141)(114 142)(115 143)(116 144)(117 133)(118 134)(119 135)(120 136)(157 188)(158 189)(159 190)(160 191)(161 192)(162 181)(163 182)(164 183)(165 184)(166 185)(167 186)(168 187)
(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)
(1 123 150 57)(2 124 151 58)(3 125 152 59)(4 126 153 60)(5 127 154 49)(6 128 155 50)(7 129 156 51)(8 130 145 52)(9 131 146 53)(10 132 147 54)(11 121 148 55)(12 122 149 56)(13 137 175 163)(14 138 176 164)(15 139 177 165)(16 140 178 166)(17 141 179 167)(18 142 180 168)(19 143 169 157)(20 144 170 158)(21 133 171 159)(22 134 172 160)(23 135 173 161)(24 136 174 162)(25 91 106 83)(26 92 107 84)(27 93 108 73)(28 94 97 74)(29 95 98 75)(30 96 99 76)(31 85 100 77)(32 86 101 78)(33 87 102 79)(34 88 103 80)(35 89 104 81)(36 90 105 82)(37 120 68 181)(38 109 69 182)(39 110 70 183)(40 111 71 184)(41 112 72 185)(42 113 61 186)(43 114 62 187)(44 115 63 188)(45 116 64 189)(46 117 65 190)(47 118 66 191)(48 119 67 192)
(1 24 150 174)(2 23 151 173)(3 22 152 172)(4 21 153 171)(5 20 154 170)(6 19 155 169)(7 18 156 180)(8 17 145 179)(9 16 146 178)(10 15 147 177)(11 14 148 176)(12 13 149 175)(25 189 106 116)(26 188 107 115)(27 187 108 114)(28 186 97 113)(29 185 98 112)(30 184 99 111)(31 183 100 110)(32 182 101 109)(33 181 102 120)(34 192 103 119)(35 191 104 118)(36 190 105 117)(37 87 68 79)(38 86 69 78)(39 85 70 77)(40 96 71 76)(41 95 72 75)(42 94 61 74)(43 93 62 73)(44 92 63 84)(45 91 64 83)(46 90 65 82)(47 89 66 81)(48 88 67 80)(49 144 127 158)(50 143 128 157)(51 142 129 168)(52 141 130 167)(53 140 131 166)(54 139 132 165)(55 138 121 164)(56 137 122 163)(57 136 123 162)(58 135 124 161)(59 134 125 160)(60 133 126 159)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4L 4M ··· 4T 6A ··· 6G 12A ··· 12X order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 ··· 2 12 ··· 12 2 ··· 2 2 ··· 2

60 irreducible representations

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

Matrix representation of C2×C122Q8 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
,
 1 11 0 0 0 0 1 12 0 0 0 0 0 0 6 10 0 0 0 0 3 3 0 0 0 0 0 0 12 11 0 0 0 0 1 1
,
 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 11 0 0 0 0 1 1
,
 4 4 0 0 0 0 6 9 0 0 0 0 0 0 3 3 0 0 0 0 6 10 0 0 0 0 0 0 1 8 0 0 0 0 3 12

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],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,6,3,0,0,0,0,10,3,0,0,0,0,0,0,12,1,0,0,0,0,11,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,11,1],[4,6,0,0,0,0,4,9,0,0,0,0,0,0,3,6,0,0,0,0,3,10,0,0,0,0,0,0,1,3,0,0,0,0,8,12] >;

C2×C122Q8 in GAP, Magma, Sage, TeX

C_2\times C_{12}\rtimes_2Q_8
% in TeX

G:=Group("C2xC12:2Q8");
// GroupNames label

G:=SmallGroup(192,1027);
// by ID

G=gap.SmallGroup(192,1027);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,184,675,80,6278]);
// Polycyclic

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

׿
×
𝔽