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G = C2×C12.3Q8  order 192 = 26·3

Direct product of C2 and C12.3Q8

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C12.3Q8, C4⋊C4.260D6, C12.66(C2×Q8), (C2×C12).30Q8, C6.8(C22×Q8), (C2×C6).43C24, C6⋊3(C42.C2), (C2×C4).37Dic6, C4.31(C2×Dic6), (C22×C4).374D6, (C2×C12).134C23, C22.81(S3×C23), C2.10(C22×Dic6), C22.37(C2×Dic6), C4⋊Dic3.357C22, C23.333(C22×S3), (C22×C12).73C22, (C22×C6).392C23, C22.72(D4⋊2S3), Dic3⋊C4.104C22, C22.32(Q8⋊3S3), (C4×Dic3).248C22, (C2×Dic3).183C23, (C22×Dic3).208C22, (C6×C4⋊C4).19C2, (C2×C4⋊C4).26S3, C3⋊3(C2×C42.C2), C6.70(C2×C4○D4), (C2×C6).51(C2×Q8), C2.5(C2×Q8⋊3S3), (C2×C4×Dic3).13C2, C2.14(C2×D4⋊2S3), (C2×C4⋊Dic3).43C2, (C2×C6).170(C4○D4), (C3×C4⋊C4).292C22, (C2×Dic3⋊C4).22C2, (C2×C4).139(C22×S3), SmallGroup(192,1058)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C2×C12.3Q8
Chief series
C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×C4×Dic3 — C2×C12.3Q8
Lower central
C3 — C2×C6 — C2×C12.3Q8

Subgroups: 440 in 226 conjugacy classes, 127 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×12], C22, C22 [×6], C6 [×3], C6 [×4], C2×C4 [×10], C2×C4 [×20], C23, Dic3 [×8], C12 [×4], C12 [×4], C2×C6, C2×C6 [×6], C42 [×4], C4⋊C4 [×4], C4⋊C4 [×20], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Dic3 [×8], C2×Dic3 [×8], C2×C12 [×10], C2×C12 [×4], C22×C6, C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×5], C42.C2 [×8], C4×Dic3 [×4], Dic3⋊C4 [×8], C4⋊Dic3 [×12], C3×C4⋊C4 [×4], C22×Dic3 [×2], C22×Dic3 [×2], C22×C12, C22×C12 [×2], C2×C42.C2, C12.3Q8 [×8], C2×C4×Dic3, C2×Dic3⋊C4 [×2], C2×C4⋊Dic3, C2×C4⋊Dic3 [×2], C6×C4⋊C4, C2×C12.3Q8

Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×4], C24, Dic6 [×4], C22×S3 [×7], C42.C2 [×4], C22×Q8, C2×C4○D4 [×2], C2×Dic6 [×6], D4⋊2S3 [×2], Q8⋊3S3 [×2], S3×C23, C2×C42.C2, C12.3Q8 [×4], C22×Dic6, C2×D4⋊2S3, C2×Q8⋊3S3, C2×C12.3Q8

Generators and relations
 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=b6c-1 >

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

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

100000
010000
0012000
0001200
000010
000001
,
400000
3100000
003000
0012900
000064
000077
,
800000
550000
001000
000100
00001211
000001
,
120000
12120000
00111200
003200
000096
000044

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],[4,3,0,0,0,0,0,10,0,0,0,0,0,0,3,12,0,0,0,0,0,9,0,0,0,0,0,0,6,7,0,0,0,0,4,7],[8,5,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,11,1],[1,12,0,0,0,0,2,12,0,0,0,0,0,0,11,3,0,0,0,0,12,2,0,0,0,0,0,0,9,4,0,0,0,0,6,4] >;
 

48 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E4F4G4H4I···4P4Q4R4S4T6A···6G12A···12L
order12···23444444444···444446···612···12
size11···12222244446···6121212122···24···4

48 irreducible representations

dim11111122222244
type+++++++-++--+
imageC1C2C2C2C2C2S3Q8D6D6C4○D4Dic6D4⋊2S3Q8⋊3S3
kernelC2×C12.3Q8C12.3Q8C2×C4×Dic3C2×Dic3⋊C4C2×C4⋊Dic3C6×C4⋊C4C2×C4⋊C4C2×C12C4⋊C4C22×C4C2×C6C2×C4C22C22
# reps18123114438822

In GAP, Magma, Sage, TeX 

C_2\times C_{12}._3Q_8
 
% in TeX
 
G:=Group("C2xC12.3Q8");
 
// GroupNames label
 
G:=SmallGroup(192,1058);
 
// by ID
 
G=gap.SmallGroup(192,1058);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,184,1571,297,80,6278]);
 
// 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=b^6*c^-1>;
 
// generators/relations
 

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