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## G = C42.279D6order 192 = 26·3

### 2nd central extension by C42 of D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C42.279D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×C12 — C4×C3⋊C8 — C42.279D6
 Lower central C3 — C6 — C42.279D6
 Upper central C1 — C42 — C4×C8

Generators and relations for C42.279D6
G = < a,b,c,d | a4=b4=1, c6=b-1, d2=a-1b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2b2c5 >

Subgroups: 104 in 66 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C3, C4, C22, C6, C8, C2×C4, C12, C2×C6, C42, C2×C8, C2×C8, C3⋊C8, C3⋊C8, C24, C2×C12, C4×C8, C4×C8, C2×C3⋊C8, C4×C12, C2×C24, C8⋊C8, C4×C3⋊C8, C4×C24, C42.279D6
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C42, C2×C8, M4(2), C4×S3, C2×Dic3, C4×C8, C8⋊C4, S3×C8, C8⋊S3, C4.Dic3, C4×Dic3, C8⋊C8, C42.S3, C8×Dic3, C24⋊C4, C42.279D6

Smallest permutation representation of C42.279D6
Regular action on 192 points
Generators in S192
```(1 80 59 34)(2 81 60 35)(3 82 61 36)(4 83 62 37)(5 84 63 38)(6 85 64 39)(7 86 65 40)(8 87 66 41)(9 88 67 42)(10 89 68 43)(11 90 69 44)(12 91 70 45)(13 92 71 46)(14 93 72 47)(15 94 49 48)(16 95 50 25)(17 96 51 26)(18 73 52 27)(19 74 53 28)(20 75 54 29)(21 76 55 30)(22 77 56 31)(23 78 57 32)(24 79 58 33)(97 129 156 183)(98 130 157 184)(99 131 158 185)(100 132 159 186)(101 133 160 187)(102 134 161 188)(103 135 162 189)(104 136 163 190)(105 137 164 191)(106 138 165 192)(107 139 166 169)(108 140 167 170)(109 141 168 171)(110 142 145 172)(111 143 146 173)(112 144 147 174)(113 121 148 175)(114 122 149 176)(115 123 150 177)(116 124 151 178)(117 125 152 179)(118 126 153 180)(119 127 154 181)(120 128 155 182)
(1 19 13 7)(2 20 14 8)(3 21 15 9)(4 22 16 10)(5 23 17 11)(6 24 18 12)(25 43 37 31)(26 44 38 32)(27 45 39 33)(28 46 40 34)(29 47 41 35)(30 48 42 36)(49 67 61 55)(50 68 62 56)(51 69 63 57)(52 70 64 58)(53 71 65 59)(54 72 66 60)(73 91 85 79)(74 92 86 80)(75 93 87 81)(76 94 88 82)(77 95 89 83)(78 96 90 84)(97 115 109 103)(98 116 110 104)(99 117 111 105)(100 118 112 106)(101 119 113 107)(102 120 114 108)(121 139 133 127)(122 140 134 128)(123 141 135 129)(124 142 136 130)(125 143 137 131)(126 144 138 132)(145 163 157 151)(146 164 158 152)(147 165 159 153)(148 166 160 154)(149 167 161 155)(150 168 162 156)(169 187 181 175)(170 188 182 176)(171 189 183 177)(172 190 184 178)(173 191 185 179)(174 192 186 180)
(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)
(1 185 28 152 71 143 86 105)(2 124 29 110 72 190 87 157)(3 171 30 162 49 129 88 115)(4 134 31 120 50 176 89 167)(5 181 32 148 51 139 90 101)(6 144 33 106 52 186 91 153)(7 191 34 158 53 125 92 111)(8 130 35 116 54 172 93 163)(9 177 36 168 55 135 94 97)(10 140 37 102 56 182 95 149)(11 187 38 154 57 121 96 107)(12 126 39 112 58 192 73 159)(13 173 40 164 59 131 74 117)(14 136 41 98 60 178 75 145)(15 183 42 150 61 141 76 103)(16 122 43 108 62 188 77 155)(17 169 44 160 63 127 78 113)(18 132 45 118 64 174 79 165)(19 179 46 146 65 137 80 99)(20 142 47 104 66 184 81 151)(21 189 48 156 67 123 82 109)(22 128 25 114 68 170 83 161)(23 175 26 166 69 133 84 119)(24 138 27 100 70 180 85 147)```

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3 4A ··· 4L 6A 6B 6C 8A ··· 8H 8I ··· 8X 12A ··· 12L 24A ··· 24P order 1 2 2 2 3 4 ··· 4 6 6 6 8 ··· 8 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 1 1 2 1 ··· 1 2 2 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C4 C4 C8 S3 D6 Dic3 M4(2) C4×S3 S3×C8 C8⋊S3 C4.Dic3 kernel C42.279D6 C4×C3⋊C8 C4×C24 C2×C3⋊C8 C2×C24 C3⋊C8 C4×C8 C42 C2×C8 C12 C2×C4 C4 C4 C4 # reps 1 2 1 8 4 16 1 1 2 8 4 8 8 8

Matrix representation of C42.279D6 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 27 0 0 0 0 27
,
 27 0 0 0 0 27 0 0 0 0 27 0 0 0 0 27
,
 0 63 0 0 10 63 0 0 0 0 72 71 0 0 60 1
,
 34 20 0 0 54 39 0 0 0 0 65 69 0 0 34 8
`G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,27,0,0,0,0,27],[27,0,0,0,0,27,0,0,0,0,27,0,0,0,0,27],[0,10,0,0,63,63,0,0,0,0,72,60,0,0,71,1],[34,54,0,0,20,39,0,0,0,0,65,34,0,0,69,8] >;`

C42.279D6 in GAP, Magma, Sage, TeX

`C_4^2._{279}D_6`
`% in TeX`

`G:=Group("C4^2.279D6");`
`// GroupNames label`

`G:=SmallGroup(192,13);`
`// by ID`

`G=gap.SmallGroup(192,13);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,477,64,184,80,6278]);`
`// Polycyclic`

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

׿
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