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## G = (C22×C4).85D6order 192 = 26·3

### 69th non-split extension by C22×C4 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C22×C4).85D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×Dic3⋊C4 — (C22×C4).85D6
 Lower central C3 — C22×C6 — (C22×C4).85D6
 Upper central C1 — C23 — C2.C42

Generators and relations for (C22×C4).85D6
G = < a,b,c,d | a6=b4=c4=1, d2=a3c2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a3b, dbd-1=b-1, dcd-1=a3c-1 >

Subgroups: 368 in 150 conjugacy classes, 65 normal (25 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2.C42, C2×C4⋊C4, Dic3⋊C4, C4⋊Dic3, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C23.81C23, C6.C42, C3×C2.C42, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×C4⋊Dic3, (C22×C4).85D6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, D12, C22×S3, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C2×Dic6, C2×D12, S3×D4, D42S3, Q83S3, C23.81C23, C122Q8, Dic3.D4, C23.21D6, C4.Dic6, C12⋊D4, (C22×C4).85D6

Smallest permutation representation of (C22×C4).85D6
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 79 19 85)(2 80 20 86)(3 81 21 87)(4 82 22 88)(5 83 23 89)(6 84 24 90)(7 73 13 91)(8 74 14 92)(9 75 15 93)(10 76 16 94)(11 77 17 95)(12 78 18 96)(25 67 43 49)(26 68 44 50)(27 69 45 51)(28 70 46 52)(29 71 47 53)(30 72 48 54)(31 61 37 55)(32 62 38 56)(33 63 39 57)(34 64 40 58)(35 65 41 59)(36 66 42 60)(97 181 115 175)(98 182 116 176)(99 183 117 177)(100 184 118 178)(101 185 119 179)(102 186 120 180)(103 187 109 169)(104 188 110 170)(105 189 111 171)(106 190 112 172)(107 191 113 173)(108 192 114 174)(121 145 139 163)(122 146 140 164)(123 147 141 165)(124 148 142 166)(125 149 143 167)(126 150 144 168)(127 151 133 157)(128 152 134 158)(129 153 135 159)(130 154 136 160)(131 155 137 161)(132 156 138 162)
(1 58 10 49)(2 59 11 50)(3 60 12 51)(4 55 7 52)(5 56 8 53)(6 57 9 54)(13 70 22 61)(14 71 23 62)(15 72 24 63)(16 67 19 64)(17 68 20 65)(18 69 21 66)(25 82 34 73)(26 83 35 74)(27 84 36 75)(28 79 31 76)(29 80 32 77)(30 81 33 78)(37 94 46 85)(38 95 47 86)(39 96 48 87)(40 91 43 88)(41 92 44 89)(42 93 45 90)(97 148 106 151)(98 149 107 152)(99 150 108 153)(100 145 103 154)(101 146 104 155)(102 147 105 156)(109 160 118 163)(110 161 119 164)(111 162 120 165)(112 157 115 166)(113 158 116 167)(114 159 117 168)(121 172 130 175)(122 173 131 176)(123 174 132 177)(124 169 127 178)(125 170 128 179)(126 171 129 180)(133 184 142 187)(134 185 143 188)(135 186 144 189)(136 181 139 190)(137 182 140 191)(138 183 141 192)
(1 107 7 101)(2 106 8 100)(3 105 9 99)(4 104 10 98)(5 103 11 97)(6 108 12 102)(13 119 19 113)(14 118 20 112)(15 117 21 111)(16 116 22 110)(17 115 23 109)(18 120 24 114)(25 131 31 125)(26 130 32 124)(27 129 33 123)(28 128 34 122)(29 127 35 121)(30 132 36 126)(37 143 43 137)(38 142 44 136)(39 141 45 135)(40 140 46 134)(41 139 47 133)(42 144 48 138)(49 155 55 149)(50 154 56 148)(51 153 57 147)(52 152 58 146)(53 151 59 145)(54 156 60 150)(61 167 67 161)(62 166 68 160)(63 165 69 159)(64 164 70 158)(65 163 71 157)(66 168 72 162)(73 179 79 173)(74 178 80 172)(75 177 81 171)(76 176 82 170)(77 175 83 169)(78 180 84 174)(85 191 91 185)(86 190 92 184)(87 189 93 183)(88 188 94 182)(89 187 95 181)(90 192 96 186)

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4F 4G ··· 4N 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 4 ··· 4 12 ··· 12 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - + - + + - + image C1 C2 C2 C2 C2 S3 D4 D4 Q8 D6 C4○D4 Dic6 D12 S3×D4 D4⋊2S3 Q8⋊3S3 kernel (C22×C4).85D6 C6.C42 C3×C2.C42 C2×Dic3⋊C4 C2×C4⋊Dic3 C2.C42 C2×Dic3 C2×C12 C2×C12 C22×C4 C2×C6 C2×C4 C2×C4 C22 C22 C22 # reps 1 2 1 1 3 1 2 2 4 3 6 8 4 1 2 1

Matrix representation of (C22×C4).85D6 in GL6(𝔽13)

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

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

(C22×C4).85D6 in GAP, Magma, Sage, TeX

(C_2^2\times C_4)._{85}D_6
% in TeX

G:=Group("(C2^2xC4).85D6");
// GroupNames label

G:=SmallGroup(192,220);
// by ID

G=gap.SmallGroup(192,220);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,344,254,387,58,6278]);
// Polycyclic

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

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