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## G = Q8×C2×C12order 192 = 26·3

### Direct product of C2×C12 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q8×C2×C12
G = < a,b,c,d | a2=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 322 in 298 conjugacy classes, 274 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×C12, C2×C12, C3×Q8, C22×C6, C2×C42, C2×C4⋊C4, C4×Q8, C22×Q8, C4×C12, C3×C4⋊C4, C22×C12, C22×C12, C6×Q8, C2×C4×Q8, C2×C4×C12, C6×C4⋊C4, Q8×C12, Q8×C2×C6, Q8×C2×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, Q8, C23, C12, C2×C6, C22×C4, C2×Q8, C4○D4, C24, C2×C12, C3×Q8, C22×C6, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, C22×C12, C6×Q8, C3×C4○D4, C23×C6, C2×C4×Q8, Q8×C12, C23×C12, Q8×C2×C6, C6×C4○D4, Q8×C2×C12

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

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

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

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

120 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4H 4I ··· 4AF 6A ··· 6N 12A ··· 12P 12Q ··· 12BL order 1 2 ··· 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 ··· 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 Q8 C4○D4 C3×Q8 C3×C4○D4 kernel Q8×C2×C12 C2×C4×C12 C6×C4⋊C4 Q8×C12 Q8×C2×C6 C2×C4×Q8 C6×Q8 C2×C42 C2×C4⋊C4 C4×Q8 C22×Q8 C2×Q8 C2×C12 C2×C6 C2×C4 C22 # reps 1 3 3 8 1 2 16 6 6 16 2 32 4 4 8 8

Matrix representation of Q8×C2×C12 in GL4(𝔽13) generated by

 12 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 10 0 0 0 0 8 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 12 0 0 0 0 0 1 0 0 12 0
,
 12 0 0 0 0 12 0 0 0 0 3 9 0 0 9 10
G:=sub<GL(4,GF(13))| [12,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[10,0,0,0,0,8,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0],[12,0,0,0,0,12,0,0,0,0,3,9,0,0,9,10] >;

Q8×C2×C12 in GAP, Magma, Sage, TeX

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

G:=Group("Q8xC2xC12");
// GroupNames label

G:=SmallGroup(192,1405);
// by ID

G=gap.SmallGroup(192,1405);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,344,772]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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