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## G = C6×C4⋊Q8order 192 = 26·3

### Direct product of C6 and C4⋊Q8

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C6×C4⋊Q8
G = < a,b,c,d | a6=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 370 in 290 conjugacy classes, 210 normal (14 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×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, C6×Q8, C2×C4⋊Q8, C2×C4×C12, C6×C4⋊C4, C3×C4⋊Q8, Q8×C2×C6, C6×C4⋊Q8
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, C2×D4, C2×Q8, C24, C3×D4, C3×Q8, C22×C6, C4⋊Q8, C22×D4, C22×Q8, C6×D4, C6×Q8, C23×C6, C2×C4⋊Q8, C3×C4⋊Q8, D4×C2×C6, Q8×C2×C6, C6×C4⋊Q8

Smallest permutation representation of C6×C4⋊Q8
Regular action on 192 points
Generators in S192
(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)
(1 104 26 92)(2 105 27 93)(3 106 28 94)(4 107 29 95)(5 108 30 96)(6 103 25 91)(7 121 187 133)(8 122 188 134)(9 123 189 135)(10 124 190 136)(11 125 191 137)(12 126 192 138)(13 115 20 127)(14 116 21 128)(15 117 22 129)(16 118 23 130)(17 119 24 131)(18 120 19 132)(31 97 42 109)(32 98 37 110)(33 99 38 111)(34 100 39 112)(35 101 40 113)(36 102 41 114)(43 79 55 67)(44 80 56 68)(45 81 57 69)(46 82 58 70)(47 83 59 71)(48 84 60 72)(49 85 61 73)(50 86 62 74)(51 87 63 75)(52 88 64 76)(53 89 65 77)(54 90 66 78)(139 163 151 175)(140 164 152 176)(141 165 153 177)(142 166 154 178)(143 167 155 179)(144 168 156 180)(145 169 157 181)(146 170 158 182)(147 171 159 183)(148 172 160 184)(149 173 161 185)(150 174 162 186)
(1 71 32 89)(2 72 33 90)(3 67 34 85)(4 68 35 86)(5 69 36 87)(6 70 31 88)(7 160 20 142)(8 161 21 143)(9 162 22 144)(10 157 23 139)(11 158 24 140)(12 159 19 141)(13 154 187 148)(14 155 188 149)(15 156 189 150)(16 151 190 145)(17 152 191 146)(18 153 192 147)(25 82 42 76)(26 83 37 77)(27 84 38 78)(28 79 39 73)(29 80 40 74)(30 81 41 75)(43 100 61 106)(44 101 62 107)(45 102 63 108)(46 97 64 103)(47 98 65 104)(48 99 66 105)(49 94 55 112)(50 95 56 113)(51 96 57 114)(52 91 58 109)(53 92 59 110)(54 93 60 111)(115 178 133 172)(116 179 134 173)(117 180 135 174)(118 175 136 169)(119 176 137 170)(120 177 138 171)(121 184 127 166)(122 185 128 167)(123 186 129 168)(124 181 130 163)(125 182 131 164)(126 183 132 165)
(1 119 32 137)(2 120 33 138)(3 115 34 133)(4 116 35 134)(5 117 36 135)(6 118 31 136)(7 94 20 112)(8 95 21 113)(9 96 22 114)(10 91 23 109)(11 92 24 110)(12 93 19 111)(13 100 187 106)(14 101 188 107)(15 102 189 108)(16 97 190 103)(17 98 191 104)(18 99 192 105)(25 130 42 124)(26 131 37 125)(27 132 38 126)(28 127 39 121)(29 128 40 122)(30 129 41 123)(43 148 61 154)(44 149 62 155)(45 150 63 156)(46 145 64 151)(47 146 65 152)(48 147 66 153)(49 142 55 160)(50 143 56 161)(51 144 57 162)(52 139 58 157)(53 140 59 158)(54 141 60 159)(67 172 85 178)(68 173 86 179)(69 174 87 180)(70 169 88 175)(71 170 89 176)(72 171 90 177)(73 166 79 184)(74 167 80 185)(75 168 81 186)(76 163 82 181)(77 164 83 182)(78 165 84 183)

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4L 4M ··· 4T 6A ··· 6N 12A ··· 12X 12Y ··· 12AN order 1 2 ··· 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 ··· 1 1 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4

84 irreducible representations

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

Matrix representation of C6×C4⋊Q8 in GL6(𝔽13)

 3 0 0 0 0 0 0 3 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 6 11 0 0 0 0 12 7
,
 9 3 0 0 0 0 3 4 0 0 0 0 0 0 4 3 0 0 0 0 3 9 0 0 0 0 0 0 1 7 0 0 0 0 9 12

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,6,12,0,0,0,0,11,7],[9,3,0,0,0,0,3,4,0,0,0,0,0,0,4,3,0,0,0,0,3,9,0,0,0,0,0,0,1,9,0,0,0,0,7,12] >;

C6×C4⋊Q8 in GAP, Magma, Sage, TeX

C_6\times C_4\rtimes Q_8
% in TeX

G:=Group("C6xC4:Q8");
// GroupNames label

G:=SmallGroup(192,1420);
// by ID

G=gap.SmallGroup(192,1420);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,701,344,2102,520]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^4=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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