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G = C2×C12⋊Q8  order 192 = 26·3

Direct product of C2 and C12⋊Q8

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

Aliases: C2×C12⋊Q8, C6⋊2(C4⋊Q8), C12⋊4(C2×Q8), (C2×C12)⋊2Q8, C4⋊2(C2×Dic6), (C2×C4)⋊6Dic6, C4⋊C4.258D6, (C2×Dic3)⋊8Q8, Dic3⋊1(C2×Q8), C6.7(C22×Q8), (C2×C6).41C24, Dic3.2(C2×D4), C6.39(C22×D4), C22.30(S3×Q8), (C22×C4).189D6, C22.129(S3×D4), C2.9(C22×Dic6), (C2×C12).132C23, (C2×Dic3).120D4, C22.79(S3×C23), (C22×Dic6).8C2, C22.36(C2×Dic6), C4⋊Dic3.356C22, C23.331(C22×S3), (C22×C6).390C23, (C2×Dic3).13C23, Dic3⋊C4.102C22, (C22×C12).214C22, (C2×Dic6).226C22, (C4×Dic3).247C22, (C22×Dic3).206C22, C3⋊2(C2×C4⋊Q8), C2.5(C2×S3×Q8), C2.13(C2×S3×D4), (C6×C4⋊C4).17C2, (C2×C4⋊C4).24S3, (C2×C6).50(C2×Q8), (C2×C6).385(C2×D4), (C2×C4×Dic3).12C2, (C2×C4⋊Dic3).42C2, (C3×C4⋊C4).290C22, (C2×Dic3⋊C4).21C2, (C2×C4).137(C22×S3), SmallGroup(192,1056)

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

Subgroups: 632 in 290 conjugacy classes, 143 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×16], C22, C22 [×6], C6 [×3], C6 [×4], C2×C4 [×10], C2×C4 [×24], Q8 [×16], C23, Dic3 [×8], Dic3 [×4], C12 [×4], C12 [×4], C2×C6, C2×C6 [×6], C42 [×4], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×16], Dic6 [×16], C2×Dic3 [×16], C2×Dic3 [×4], C2×C12 [×10], C2×C12 [×4], C22×C6, C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×3], C4⋊Q8 [×8], C22×Q8 [×2], C4×Dic3 [×4], Dic3⋊C4 [×8], C4⋊Dic3 [×4], C3×C4⋊C4 [×4], C2×Dic6 [×8], C2×Dic6 [×8], C22×Dic3 [×2], C22×Dic3 [×2], C22×C12, C22×C12 [×2], C2×C4⋊Q8, C12⋊Q8 [×8], C2×C4×Dic3, C2×Dic3⋊C4 [×2], C2×C4⋊Dic3, C6×C4⋊C4, C22×Dic6 [×2], C2×C12⋊Q8

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

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=c-1 >

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

(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)

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

100000
010000
0012000
0001200
0000120
0000012
,
1060000
730000
00121100
001100
00001212
000010
,
6100000
370000
00121100
000100
000036
0000710
,
100000
010000
001000
000100
000005
000050

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] >;
 

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

dim1111111222222244
type+++++++++--++-+-
imageC1C2C2C2C2C2C2S3D4Q8Q8D6D6Dic6S3×D4S3×Q8
kernelC2×C12⋊Q8C12⋊Q8C2×C4×Dic3C2×Dic3⋊C4C2×C4⋊Dic3C6×C4⋊C4C22×Dic6C2×C4⋊C4C2×Dic3C2×Dic3C2×C12C4⋊C4C22×C4C2×C4C22C22
# reps1812112144443822

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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