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## G = C2×C12⋊Q8order 192 = 26·3

### Direct product of C2 and C12⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C12⋊Q8
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×C4×Dic3 — C2×C12⋊Q8
 Lower central C3 — C2×C6 — C2×C12⋊Q8
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C2×C12⋊Q8
G = < a,b,c,d | a2=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=c-1 >

Subgroups: 632 in 290 conjugacy classes, 143 normal (21 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, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4⋊Q8, C22×Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×Dic6, C2×Dic6, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C2×C4⋊Q8, C12⋊Q8, C2×C4×Dic3, C2×Dic3⋊C4, C2×C4⋊Dic3, C6×C4⋊C4, C22×Dic6, C2×C12⋊Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C24, Dic6, C22×S3, C4⋊Q8, C22×D4, C22×Q8, C2×Dic6, S3×D4, S3×Q8, S3×C23, C2×C4⋊Q8, C12⋊Q8, C22×Dic6, C2×S3×D4, C2×S3×Q8, C2×C12⋊Q8

Smallest permutation representation of C2×C12⋊Q8
Regular action on 192 points
Generators in S192
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 25)(13 104)(14 105)(15 106)(16 107)(17 108)(18 97)(19 98)(20 99)(21 100)(22 101)(23 102)(24 103)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 49)(46 50)(47 51)(48 52)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 109)(72 110)(73 170)(74 171)(75 172)(76 173)(77 174)(78 175)(79 176)(80 177)(81 178)(82 179)(83 180)(84 169)(85 130)(86 131)(87 132)(88 121)(89 122)(90 123)(91 124)(92 125)(93 126)(94 127)(95 128)(96 129)(133 157)(134 158)(135 159)(136 160)(137 161)(138 162)(139 163)(140 164)(141 165)(142 166)(143 167)(144 168)(145 189)(146 190)(147 191)(148 192)(149 181)(150 182)(151 183)(152 184)(153 185)(154 186)(155 187)(156 188)
(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 81 144 13)(2 76 133 20)(3 83 134 15)(4 78 135 22)(5 73 136 17)(6 80 137 24)(7 75 138 19)(8 82 139 14)(9 77 140 21)(10 84 141 16)(11 79 142 23)(12 74 143 18)(25 171 167 97)(26 178 168 104)(27 173 157 99)(28 180 158 106)(29 175 159 101)(30 170 160 108)(31 177 161 103)(32 172 162 98)(33 179 163 105)(34 174 164 100)(35 169 165 107)(36 176 166 102)(37 71 146 85)(38 66 147 92)(39 61 148 87)(40 68 149 94)(41 63 150 89)(42 70 151 96)(43 65 152 91)(44 72 153 86)(45 67 154 93)(46 62 155 88)(47 69 156 95)(48 64 145 90)(49 117 186 126)(50 112 187 121)(51 119 188 128)(52 114 189 123)(53 109 190 130)(54 116 191 125)(55 111 192 132)(56 118 181 127)(57 113 182 122)(58 120 183 129)(59 115 184 124)(60 110 185 131)
(1 120 144 129)(2 113 133 122)(3 118 134 127)(4 111 135 132)(5 116 136 125)(6 109 137 130)(7 114 138 123)(8 119 139 128)(9 112 140 121)(10 117 141 126)(11 110 142 131)(12 115 143 124)(13 183 81 58)(14 188 82 51)(15 181 83 56)(16 186 84 49)(17 191 73 54)(18 184 74 59)(19 189 75 52)(20 182 76 57)(21 187 77 50)(22 192 78 55)(23 185 79 60)(24 190 80 53)(25 65 167 91)(26 70 168 96)(27 63 157 89)(28 68 158 94)(29 61 159 87)(30 66 160 92)(31 71 161 85)(32 64 162 90)(33 69 163 95)(34 62 164 88)(35 67 165 93)(36 72 166 86)(37 103 146 177)(38 108 147 170)(39 101 148 175)(40 106 149 180)(41 99 150 173)(42 104 151 178)(43 97 152 171)(44 102 153 176)(45 107 154 169)(46 100 155 174)(47 105 156 179)(48 98 145 172)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 4S 4T 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 3 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 4 4 4 4 6 ··· 6 12 12 12 12 2 ··· 2 4 ··· 4

48 irreducible representations

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

Matrix representation of C2×C12⋊Q8 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 12 0 0 0 0 0 0 12
,
 10 6 0 0 0 0 7 3 0 0 0 0 0 0 12 11 0 0 0 0 1 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0
,
 6 10 0 0 0 0 3 7 0 0 0 0 0 0 12 11 0 0 0 0 0 1 0 0 0 0 0 0 3 6 0 0 0 0 7 10
,
 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 0 5 0 0 0 0 5 0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[10,7,0,0,0,0,6,3,0,0,0,0,0,0,12,1,0,0,0,0,11,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[6,3,0,0,0,0,10,7,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,3,7,0,0,0,0,6,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,5,0] >;

C2×C12⋊Q8 in GAP, Magma, Sage, TeX

C_2\times C_{12}\rtimes Q_8
% in TeX

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

G:=SmallGroup(192,1056);
// by ID

G=gap.SmallGroup(192,1056);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,184,675,297,80,6278]);
// 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,c*b*c^-1=b^7,d*b*d^-1=b^5,d*c*d^-1=c^-1>;
// generators/relations

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