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G = C2×C12⋊2Q8  order 192 = 26·3

Direct product of C2 and C12⋊2Q8

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

Aliases: C2×C12⋊2Q8, C42.272D6, C6⋊1(C4⋊Q8), C12⋊5(C2×Q8), (C2×C12)⋊12Q8, (C2×C4)⋊9Dic6, C4⋊3(C2×Dic6), C4.42(C2×D12), (C2×C4).97D12, C6.1(C22×D4), (C2×C12).388D4, C12.285(C2×D4), C6.2(C22×Q8), (C2×C42).20S3, (C2×C6).12C24, C2.4(C22×D12), (C22×C4).450D6, C22.62(C2×D12), C2.4(C22×Dic6), (C4×C12).312C22, (C2×C12).778C23, C22.59(S3×C23), (C2×Dic3).1C23, (C22×Dic6).7C2, C22.34(C2×Dic6), C4⋊Dic3.286C22, C23.320(C22×S3), (C22×C6).374C23, (C22×C12).521C22, (C2×Dic6).222C22, (C22×Dic3).72C22, C3⋊1(C2×C4⋊Q8), (C2×C4×C12).13C2, (C2×C6).46(C2×Q8), (C2×C6).168(C2×D4), (C2×C4⋊Dic3).24C2, (C2×C4).727(C22×S3), SmallGroup(192,1027)

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

Subgroups: 632 in 290 conjugacy classes, 159 normal (13 characteristic)
C1, C2, C2 [×6], C3, C4 [×12], C4 [×8], C22, C22 [×6], C6, C6 [×6], C2×C4 [×18], C2×C4 [×16], Q8 [×16], C23, Dic3 [×8], C12 [×12], C2×C6, C2×C6 [×6], C42 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×16], Dic6 [×16], C2×Dic3 [×8], C2×Dic3 [×8], C2×C12 [×18], C22×C6, C2×C42, C2×C4⋊C4 [×4], C4⋊Q8 [×8], C22×Q8 [×2], C4⋊Dic3 [×16], C4×C12 [×4], C2×Dic6 [×8], C2×Dic6 [×8], C22×Dic3 [×4], C22×C12, C22×C12 [×2], C2×C4⋊Q8, C12⋊2Q8 [×8], C2×C4⋊Dic3 [×4], C2×C4×C12, C22×Dic6 [×2], C2×C12⋊2Q8

Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], Q8 [×8], C23 [×15], D6 [×7], C2×D4 [×6], C2×Q8 [×12], C24, Dic6 [×8], D12 [×4], C22×S3 [×7], C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C2×Dic6 [×12], C2×D12 [×6], S3×C23, C2×C4⋊Q8, C12⋊2Q8 [×4], C22×Dic6 [×2], C22×D12, C2×C12⋊2Q8

Generators and relations
 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 >

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

(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)

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

100000
010000
0012000
0001200
000010
000001
,
1110000
1120000
0061000
003300
00001211
000011
,
100000
010000
001000
000100
00001211
000011
,
440000
690000
003300
0061000
000018
0000312

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] >;
 

60 conjugacy classes

class 1 2A···2G 3 4A···4L4M···4T6A···6G12A···12X
order12···234···44···46···612···12
size11···122···212···122···22···2

60 irreducible representations

dim111112222222
type+++++++-++-+
imageC1C2C2C2C2S3D4Q8D6D6Dic6D12
kernelC2×C12⋊2Q8C12⋊2Q8C2×C4⋊Dic3C2×C4×C12C22×Dic6C2×C42C2×C12C2×C12C42C22×C4C2×C4C2×C4
# reps1841214843168

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
 

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