Copied to
clipboard

## G = C42.14D6order 192 = 26·3

### 14th non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42.14D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C12.6Q8 — C42.14D6
 Lower central C3 — C6 — C2×C12 — C42.14D6
 Upper central C1 — C22 — C42 — C8⋊C4

Generators and relations for C42.14D6
G = < a,b,c,d | a4=b4=1, c6=a2b, d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b-1c5 >

Subgroups: 248 in 90 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C2×C8, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C8⋊C4, Q8⋊C4, C42.C2, C4⋊Q8, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C42.30C22, C2.Dic12, C3×C8⋊C4, C122Q8, C12.6Q8, C42.14D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4.4D4, C8.C22, C2×D12, C4○D12, C42.30C22, C427S3, C8.D6, C42.14D6

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

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - - image C1 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 D12 C4○D12 C8.C22 C8.D6 kernel C42.14D6 C2.Dic12 C3×C8⋊C4 C12⋊2Q8 C12.6Q8 C8⋊C4 C2×C12 C42 C2×C8 C12 C2×C4 C4 C6 C2 # reps 1 4 1 1 1 1 2 1 2 4 4 8 2 4

Matrix representation of C42.14D6 in GL6(𝔽73)

 52 2 0 0 0 0 71 21 0 0 0 0 0 0 26 34 22 70 0 0 39 65 3 25 0 0 50 37 47 39 0 0 36 13 34 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 14 0 0 0 0 59 66 0 0 0 0 0 0 7 14 0 0 0 0 59 66
,
 46 0 0 0 0 0 0 46 0 0 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 66 7 0 0 0 0 66 59 0 0
,
 15 19 0 0 0 0 42 58 0 0 0 0 0 0 38 13 46 63 0 0 48 35 17 27 0 0 46 63 11 3 0 0 17 27 65 62

`G:=sub<GL(6,GF(73))| [52,71,0,0,0,0,2,21,0,0,0,0,0,0,26,39,50,36,0,0,34,65,37,13,0,0,22,3,47,34,0,0,70,25,39,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,59,0,0,0,0,14,66,0,0,0,0,0,0,7,59,0,0,0,0,14,66],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,66,66,0,0,0,0,7,59,0,0,72,1,0,0,0,0,72,0,0,0],[15,42,0,0,0,0,19,58,0,0,0,0,0,0,38,48,46,17,0,0,13,35,63,27,0,0,46,17,11,65,0,0,63,27,3,62] >;`

C42.14D6 in GAP, Magma, Sage, TeX

`C_4^2._{14}D_6`
`% in TeX`

`G:=Group("C4^2.14D6");`
`// GroupNames label`

`G:=SmallGroup(192,262);`
`// by ID`

`G=gap.SmallGroup(192,262);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,344,254,387,142,1123,136,6278]);`
`// Polycyclic`

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

׿
×
𝔽