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G = C42.215D6order 192 = 26·3

35th non-split extension by C42 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42.215D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C3⋊C8 — C4×C3⋊C8 — C42.215D6
 Lower central C3 — C6 — C2×C12 — C42.215D6
 Upper central C1 — C22 — C42 — C42.C2

Generators and relations for C42.215D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=bc5 >

Subgroups: 192 in 86 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×6], C22, C6, C6 [×2], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×4], Dic3 [×2], C12 [×2], C12 [×4], C2×C6, C42, C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×2], C3⋊C8 [×4], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C2×C12 [×2], C4×C8, C4.Q8 [×2], C2.D8 [×2], C42.C2, C42.C2, C2×C3⋊C8 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×2], C8.5Q8, C4×C3⋊C8, C6.Q16 [×2], C12.Q8 [×2], C12.6Q8, C3×C42.C2, C42.215D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D6 [×3], C2×D4, C2×Q8 [×2], C3⋊D4 [×2], C22×S3, C4⋊Q8, C4○D8 [×2], S3×Q8 [×2], C2×C3⋊D4, C8.5Q8, Dic3⋊Q8, Q8.13D6 [×2], C42.215D6

Smallest permutation representation of C42.215D6
Regular action on 192 points
Generators in S192
```(1 62 7 68)(2 128 8 122)(3 64 9 70)(4 130 10 124)(5 66 11 72)(6 132 12 126)(13 113 19 119)(14 162 20 168)(15 115 21 109)(16 164 22 158)(17 117 23 111)(18 166 24 160)(25 75 31 81)(26 190 32 184)(27 77 33 83)(28 192 34 186)(29 79 35 73)(30 182 36 188)(37 173 43 179)(38 144 44 138)(39 175 45 169)(40 134 46 140)(41 177 47 171)(42 136 48 142)(49 84 55 78)(50 187 56 181)(51 74 57 80)(52 189 58 183)(53 76 59 82)(54 191 60 185)(61 156 67 150)(63 146 69 152)(65 148 71 154)(85 120 91 114)(86 157 92 163)(87 110 93 116)(88 159 94 165)(89 112 95 118)(90 161 96 167)(97 139 103 133)(98 170 104 176)(99 141 105 135)(100 172 106 178)(101 143 107 137)(102 174 108 180)(121 151 127 145)(123 153 129 147)(125 155 131 149)
(1 73 145 181)(2 182 146 74)(3 75 147 183)(4 184 148 76)(5 77 149 185)(6 186 150 78)(7 79 151 187)(8 188 152 80)(9 81 153 189)(10 190 154 82)(11 83 155 191)(12 192 156 84)(13 137 96 173)(14 174 85 138)(15 139 86 175)(16 176 87 140)(17 141 88 177)(18 178 89 142)(19 143 90 179)(20 180 91 144)(21 133 92 169)(22 170 93 134)(23 135 94 171)(24 172 95 136)(25 129 58 70)(26 71 59 130)(27 131 60 72)(28 61 49 132)(29 121 50 62)(30 63 51 122)(31 123 52 64)(32 65 53 124)(33 125 54 66)(34 67 55 126)(35 127 56 68)(36 69 57 128)(37 119 107 161)(38 162 108 120)(39 109 97 163)(40 164 98 110)(41 111 99 165)(42 166 100 112)(43 113 101 167)(44 168 102 114)(45 115 103 157)(46 158 104 116)(47 117 105 159)(48 160 106 118)
(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)
(1 178 7 172)(2 94 8 88)(3 176 9 170)(4 92 10 86)(5 174 11 180)(6 90 12 96)(13 150 19 156)(14 83 20 77)(15 148 21 154)(16 81 22 75)(17 146 23 152)(18 79 24 73)(25 110 31 116)(26 45 32 39)(27 120 33 114)(28 43 34 37)(29 118 35 112)(30 41 36 47)(38 66 44 72)(40 64 46 70)(42 62 48 68)(49 101 55 107)(50 160 56 166)(51 99 57 105)(52 158 58 164)(53 97 59 103)(54 168 60 162)(61 167 67 161)(63 165 69 159)(65 163 71 157)(74 171 80 177)(76 169 82 175)(78 179 84 173)(85 191 91 185)(87 189 93 183)(89 187 95 181)(98 123 104 129)(100 121 106 127)(102 131 108 125)(109 130 115 124)(111 128 117 122)(113 126 119 132)(133 190 139 184)(134 147 140 153)(135 188 141 182)(136 145 142 151)(137 186 143 192)(138 155 144 149)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3 4A ··· 4F 4G 4H 4I 4J 6A 6B 6C 8A ··· 8H 12A ··· 12F 12G 12H 12I 12J order 1 2 2 2 3 4 ··· 4 4 4 4 4 6 6 6 8 ··· 8 12 ··· 12 12 12 12 12 size 1 1 1 1 2 2 ··· 2 8 8 24 24 2 2 2 6 ··· 6 4 ··· 4 8 8 8 8

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 S3 Q8 D4 D6 D6 C3⋊D4 C4○D8 S3×Q8 Q8.13D6 kernel C42.215D6 C4×C3⋊C8 C6.Q16 C12.Q8 C12.6Q8 C3×C42.C2 C42.C2 C3⋊C8 C2×C12 C42 C4⋊C4 C2×C4 C6 C4 C2 # reps 1 1 2 2 1 1 1 4 2 1 2 4 8 2 4

Matrix representation of C42.215D6 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 27 0 0 0 0 46 0 0 0 0 0 0 0 27 0 0 0 0 0 0 27
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 72 3 0 0 0 0 48 1
,
 30 43 0 0 0 0 30 60 0 0 0 0 0 0 26 37 0 0 0 0 37 47 0 0 0 0 0 0 51 62 0 0 0 0 64 22
,
 59 66 0 0 0 0 7 14 0 0 0 0 0 0 30 14 0 0 0 0 14 43 0 0 0 0 0 0 53 21 0 0 0 0 40 20

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,48,0,0,0,0,3,1],[30,30,0,0,0,0,43,60,0,0,0,0,0,0,26,37,0,0,0,0,37,47,0,0,0,0,0,0,51,64,0,0,0,0,62,22],[59,7,0,0,0,0,66,14,0,0,0,0,0,0,30,14,0,0,0,0,14,43,0,0,0,0,0,0,53,40,0,0,0,0,21,20] >;`

C42.215D6 in GAP, Magma, Sage, TeX

`C_4^2._{215}D_6`
`% in TeX`

`G:=Group("C4^2.215D6");`
`// GroupNames label`

`G:=SmallGroup(192,624);`
`// by ID`

`G=gap.SmallGroup(192,624);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,64,422,471,58,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^5>;`
`// generators/relations`

׿
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