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## G = C6.(C4×D4)  order 192 = 26·3

### 3rd non-split extension by C6 of C4×D4 acting via C4×D4/C22⋊C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.(C4×D4)
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4×Dic3 — C6.(C4×D4)
 Lower central C3 — C2×C6 — C6.(C4×D4)
 Upper central C1 — C23 — C2.C42

Generators and relations for C6.(C4×D4)
G = < a,b,c,d | a6=b4=c4=1, d2=a3, bab-1=cac-1=a-1, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 352 in 154 conjugacy classes, 67 normal (51 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C2×C4, C23, 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, C22×Dic3, C22×C12, C23.63C23, C6.C42, C3×C2.C42, C2×C4×Dic3, C2×Dic3⋊C4, C6.(C4×D4)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C22×C4, C2×D4, C2×Q8, C4○D4, C4×S3, C22×S3, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, S3×C2×C4, C4○D12, S3×D4, D42S3, S3×Q8, C23.63C23, C422S3, C23.8D6, Dic34D4, C23.9D6, Dic6⋊C4, Dic3.Q8, D6⋊Q8, C6.(C4×D4)

Smallest permutation representation of C6.(C4×D4)
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)
(1 119 19 104)(2 118 20 103)(3 117 21 108)(4 116 22 107)(5 115 23 106)(6 120 24 105)(7 124 29 114)(8 123 30 113)(9 122 25 112)(10 121 26 111)(11 126 27 110)(12 125 28 109)(13 102 186 90)(14 101 181 89)(15 100 182 88)(16 99 183 87)(17 98 184 86)(18 97 185 85)(31 143 43 128)(32 142 44 127)(33 141 45 132)(34 140 46 131)(35 139 47 130)(36 144 48 129)(37 147 54 137)(38 146 49 136)(39 145 50 135)(40 150 51 134)(41 149 52 133)(42 148 53 138)(55 167 67 152)(56 166 68 151)(57 165 69 156)(58 164 70 155)(59 163 71 154)(60 168 72 153)(61 171 78 161)(62 170 73 160)(63 169 74 159)(64 174 75 158)(65 173 76 157)(66 172 77 162)(79 191 91 176)(80 190 92 175)(81 189 93 180)(82 188 94 179)(83 187 95 178)(84 192 96 177)
(1 16 7 188)(2 15 8 187)(3 14 9 192)(4 13 10 191)(5 18 11 190)(6 17 12 189)(19 183 29 179)(20 182 30 178)(21 181 25 177)(22 186 26 176)(23 185 27 175)(24 184 28 180)(31 159 42 155)(32 158 37 154)(33 157 38 153)(34 162 39 152)(35 161 40 151)(36 160 41 156)(43 169 53 164)(44 174 54 163)(45 173 49 168)(46 172 50 167)(47 171 51 166)(48 170 52 165)(55 143 66 148)(56 142 61 147)(57 141 62 146)(58 140 63 145)(59 139 64 150)(60 144 65 149)(67 128 77 138)(68 127 78 137)(69 132 73 136)(70 131 74 135)(71 130 75 134)(72 129 76 133)(79 104 90 114)(80 103 85 113)(81 108 86 112)(82 107 87 111)(83 106 88 110)(84 105 89 109)(91 119 102 124)(92 118 97 123)(93 117 98 122)(94 116 99 121)(95 115 100 126)(96 120 101 125)
(1 70 4 67)(2 71 5 68)(3 72 6 69)(7 74 10 77)(8 75 11 78)(9 76 12 73)(13 138 16 135)(14 133 17 136)(15 134 18 137)(19 58 22 55)(20 59 23 56)(21 60 24 57)(25 65 28 62)(26 66 29 63)(27 61 30 64)(31 94 34 91)(32 95 35 92)(33 96 36 93)(37 100 40 97)(38 101 41 98)(39 102 42 99)(43 82 46 79)(44 83 47 80)(45 84 48 81)(49 89 52 86)(50 90 53 87)(51 85 54 88)(103 163 106 166)(104 164 107 167)(105 165 108 168)(109 170 112 173)(110 171 113 174)(111 172 114 169)(115 151 118 154)(116 152 119 155)(117 153 120 156)(121 162 124 159)(122 157 125 160)(123 158 126 161)(127 187 130 190)(128 188 131 191)(129 189 132 192)(139 175 142 178)(140 176 143 179)(141 177 144 180)(145 186 148 183)(146 181 149 184)(147 182 150 185)

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

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

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

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

Matrix representation of C6.(C4×D4) in GL5(𝔽13)

 1 0 0 0 0 0 1 1 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 12
,
 8 0 0 0 0 0 0 12 0 0 0 12 0 0 0 0 0 0 6 6 0 0 0 9 7
,
 1 0 0 0 0 0 6 3 0 0 0 10 7 0 0 0 0 0 8 11 0 0 0 0 5
,
 12 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 6 6 0 0 0 9 7

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,12,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12],[8,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,6,9,0,0,0,6,7],[1,0,0,0,0,0,6,10,0,0,0,3,7,0,0,0,0,0,8,0,0,0,0,11,5],[12,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,6,9,0,0,0,6,7] >;

C6.(C4×D4) in GAP, Magma, Sage, TeX

C_6.(C_4\times D_4)
% in TeX

G:=Group("C6.(C4xD4)");
// GroupNames label

G:=SmallGroup(192,211);
// by ID

G=gap.SmallGroup(192,211);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,64,254,219,184,6278]);
// Polycyclic

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

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