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## G = C42.264D10order 320 = 26·5

### 23rd non-split extension by C42 of D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C42.264D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C4.D20 — C42.264D10
 Lower central C5 — C10 — C2×C20 — C42.264D10
 Upper central C1 — C22 — C42 — C4×C8

Generators and relations for C42.264D10
G = < a,b,c,d | a4=b4=1, c10=a2b-1, d2=a2, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, dbd-1=b-1, dcd-1=bc9 >

Subgroups: 446 in 96 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C4×C8, D4⋊C4, Q8⋊C4, C4.4D4, C42.C2, C40, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C42.78C22, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×Dic10, C2×D20, C20.44D4, D205C4, C4×C40, C20.6Q8, C4.D20, C42.264D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C4○D8, D20, C22×D5, C42.78C22, C2×D20, C4○D20, C4.D20, D407C2, C42.264D10

Smallest permutation representation of C42.264D10
On 160 points
Generators in S160
```(1 119 50 146)(2 120 51 147)(3 81 52 148)(4 82 53 149)(5 83 54 150)(6 84 55 151)(7 85 56 152)(8 86 57 153)(9 87 58 154)(10 88 59 155)(11 89 60 156)(12 90 61 157)(13 91 62 158)(14 92 63 159)(15 93 64 160)(16 94 65 121)(17 95 66 122)(18 96 67 123)(19 97 68 124)(20 98 69 125)(21 99 70 126)(22 100 71 127)(23 101 72 128)(24 102 73 129)(25 103 74 130)(26 104 75 131)(27 105 76 132)(28 106 77 133)(29 107 78 134)(30 108 79 135)(31 109 80 136)(32 110 41 137)(33 111 42 138)(34 112 43 139)(35 113 44 140)(36 114 45 141)(37 115 46 142)(38 116 47 143)(39 117 48 144)(40 118 49 145)
(1 80 21 60)(2 41 22 61)(3 42 23 62)(4 43 24 63)(5 44 25 64)(6 45 26 65)(7 46 27 66)(8 47 28 67)(9 48 29 68)(10 49 30 69)(11 50 31 70)(12 51 32 71)(13 52 33 72)(14 53 34 73)(15 54 35 74)(16 55 36 75)(17 56 37 76)(18 57 38 77)(19 58 39 78)(20 59 40 79)(81 138 101 158)(82 139 102 159)(83 140 103 160)(84 141 104 121)(85 142 105 122)(86 143 106 123)(87 144 107 124)(88 145 108 125)(89 146 109 126)(90 147 110 127)(91 148 111 128)(92 149 112 129)(93 150 113 130)(94 151 114 131)(95 152 115 132)(96 153 116 133)(97 154 117 134)(98 155 118 135)(99 156 119 136)(100 157 120 137)
(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)
(1 10 50 59)(2 58 51 9)(3 8 52 57)(4 56 53 7)(5 6 54 55)(11 40 60 49)(12 48 61 39)(13 38 62 47)(14 46 63 37)(15 36 64 45)(16 44 65 35)(17 34 66 43)(18 42 67 33)(19 32 68 41)(20 80 69 31)(21 30 70 79)(22 78 71 29)(23 28 72 77)(24 76 73 27)(25 26 74 75)(81 133 148 106)(82 105 149 132)(83 131 150 104)(84 103 151 130)(85 129 152 102)(86 101 153 128)(87 127 154 100)(88 99 155 126)(89 125 156 98)(90 97 157 124)(91 123 158 96)(92 95 159 122)(93 121 160 94)(107 147 134 120)(108 119 135 146)(109 145 136 118)(110 117 137 144)(111 143 138 116)(112 115 139 142)(113 141 140 114)```

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

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

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

86 conjugacy classes

 class 1 2A 2B 2C 2D 4A ··· 4F 4G 4H 4I 5A 5B 8A ··· 8H 10A ··· 10F 20A ··· 20X 40A ··· 40AF order 1 2 2 2 2 4 ··· 4 4 4 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 40 2 ··· 2 40 40 40 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

86 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 C4○D8 D20 C4○D20 D40⋊7C2 kernel C42.264D10 C20.44D4 D20⋊5C4 C4×C40 C20.6Q8 C4.D20 C2×C20 C4×C8 C20 C42 C2×C8 C10 C2×C4 C4 C2 # reps 1 2 2 1 1 1 2 2 4 2 4 8 8 16 32

Matrix representation of C42.264D10 in GL4(𝔽41) generated by

 32 0 0 0 0 32 0 0 0 0 40 21 0 0 0 1
,
 30 32 0 0 9 11 0 0 0 0 32 25 0 0 0 9
,
 13 23 0 0 18 16 0 0 0 0 3 13 0 0 0 14
,
 23 13 0 0 16 18 0 0 0 0 14 28 0 0 15 27
`G:=sub<GL(4,GF(41))| [32,0,0,0,0,32,0,0,0,0,40,0,0,0,21,1],[30,9,0,0,32,11,0,0,0,0,32,0,0,0,25,9],[13,18,0,0,23,16,0,0,0,0,3,0,0,0,13,14],[23,16,0,0,13,18,0,0,0,0,14,15,0,0,28,27] >;`

C42.264D10 in GAP, Magma, Sage, TeX

`C_4^2._{264}D_{10}`
`% in TeX`

`G:=Group("C4^2.264D10");`
`// GroupNames label`

`G:=SmallGroup(320,324);`
`// by ID`

`G=gap.SmallGroup(320,324);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,142,1123,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2*b^-1,d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b*c^9>;`
`// generators/relations`

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