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## G = C6.(C4×Q8)  order 192 = 26·3

### 2nd non-split extension by C6 of C4×Q8 acting via C4×Q8/C42=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.(C4×Q8)
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×Dic3⋊C4 — C6.(C4×Q8)
 Lower central C3 — C2×C6 — C6.(C4×Q8)
 Upper central C1 — C23 — C2.C42

Generators and relations for C6.(C4×Q8)
G = < a,b,c,d | a6=b4=c4=1, d2=c2, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd-1=a3b, dcd-1=a3c-1 >

Subgroups: 384 in 170 conjugacy classes, 77 normal (51 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C22×Dic3, C22×C12, C23.65C23, C6.C42, C3×C2.C42, C2×C4×Dic3, C2×Dic3⋊C4, C2×C4⋊Dic3, C6.(C4×Q8)
Quotients:

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

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 4S 4T 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 3 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 4 4 4 4 6 ··· 6 12 12 12 12 2 ··· 2 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - - + - + - - image C1 C2 C2 C2 C2 C2 C4 S3 D4 Q8 Q8 D6 C4○D4 Dic6 C4×S3 C4○D12 S3×D4 D4⋊2S3 S3×Q8 kernel C6.(C4×Q8) C6.C42 C3×C2.C42 C2×C4×Dic3 C2×Dic3⋊C4 C2×C4⋊Dic3 Dic3⋊C4 C2.C42 C2×Dic3 C2×Dic3 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C22 C22 C22 C22 # reps 1 1 1 1 3 1 8 1 4 2 2 3 4 4 4 4 2 1 1

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

 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 0 12 0 0 0 0 1 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 2 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 12 0 0 0 0 0 5 1 0 0 0 0 0 0 12 0 0 0 0 0 12 1
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 5 2 0 0 0 0 1 8 0 0 0 0 0 0 12 0 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,2,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,12,5,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,5,1,0,0,0,0,2,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

C6.(C4×Q8) in GAP, Magma, Sage, TeX

C_6.(C_4\times Q_8)
% in TeX

G:=Group("C6.(C4xQ8)");
// GroupNames label

G:=SmallGroup(192,206);
// by ID

G=gap.SmallGroup(192,206);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,344,387,58,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^4=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^-1>;
// generators/relations

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