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## G = C42.238D10order 320 = 26·5

### 58th non-split extension by C42 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.238D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C42 — C42.238D10
 Lower central C5 — C2×C10 — C42.238D10
 Upper central C1 — C22 — C4⋊1D4

Generators and relations for C42.238D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 1038 in 310 conjugacy classes, 111 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×6], C4 [×8], C22, C22 [×16], C5, C2×C4, C2×C4 [×2], C2×C4 [×23], D4 [×20], Q8 [×4], C23 [×4], C23, D5 [×2], C10, C10 [×2], C10 [×4], C42, C42 [×3], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×7], C2×D4 [×6], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], Dic5 [×2], Dic5 [×6], C20 [×6], D10 [×2], D10 [×2], C2×C10, C2×C10 [×12], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×6], C2×Dic5 [×8], C5⋊D4 [×8], C2×C20, C2×C20 [×2], C5×D4 [×12], C22×D5, C22×C10 [×4], C22.26C24, C4×Dic5, C4×Dic5 [×2], C4⋊Dic5 [×4], C23.D5 [×8], C4×C20, C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], D42D5 [×8], C22×Dic5 [×4], C2×C5⋊D4 [×4], D4×C10 [×6], C202Q8, D5×C42, D4×Dic5 [×4], C20.17D4 [×2], C202D4 [×4], C5×C41D4, C2×D42D5 [×2], C42.238D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C22.26C24, D4×D5 [×2], D42D5 [×4], C23×D5, C2×D4×D5, C2×D42D5 [×2], C42.238D10

Smallest permutation representation of C42.238D10
On 160 points
Generators in S160
```(1 108 33 113)(2 114 34 109)(3 110 35 115)(4 116 36 101)(5 102 37 117)(6 118 38 103)(7 104 39 119)(8 120 40 105)(9 106 31 111)(10 112 32 107)(11 100 60 135)(12 136 51 91)(13 92 52 137)(14 138 53 93)(15 94 54 139)(16 140 55 95)(17 96 56 131)(18 132 57 97)(19 98 58 133)(20 134 59 99)(21 90 143 67)(22 68 144 81)(23 82 145 69)(24 70 146 83)(25 84 147 61)(26 62 148 85)(27 86 149 63)(28 64 150 87)(29 88 141 65)(30 66 142 89)(41 122 75 155)(42 156 76 123)(43 124 77 157)(44 158 78 125)(45 126 79 159)(46 160 80 127)(47 128 71 151)(48 152 72 129)(49 130 73 153)(50 154 74 121)
(1 49 11 62)(2 63 12 50)(3 41 13 64)(4 65 14 42)(5 43 15 66)(6 67 16 44)(7 45 17 68)(8 69 18 46)(9 47 19 70)(10 61 20 48)(21 140 158 118)(22 119 159 131)(23 132 160 120)(24 111 151 133)(25 134 152 112)(26 113 153 135)(27 136 154 114)(28 115 155 137)(29 138 156 116)(30 117 157 139)(31 71 58 83)(32 84 59 72)(33 73 60 85)(34 86 51 74)(35 75 52 87)(36 88 53 76)(37 77 54 89)(38 90 55 78)(39 79 56 81)(40 82 57 80)(91 121 109 149)(92 150 110 122)(93 123 101 141)(94 142 102 124)(95 125 103 143)(96 144 104 126)(97 127 105 145)(98 146 106 128)(99 129 107 147)(100 148 108 130)
(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 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 157 158 30)(22 29 159 156)(23 155 160 28)(24 27 151 154)(25 153 152 26)(31 34 58 51)(32 60 59 33)(35 40 52 57)(36 56 53 39)(37 38 54 55)(41 46 64 69)(42 68 65 45)(43 44 66 67)(47 50 70 63)(48 62 61 49)(71 74 83 86)(72 85 84 73)(75 80 87 82)(76 81 88 79)(77 78 89 90)(91 106 109 98)(92 97 110 105)(93 104 101 96)(94 95 102 103)(99 108 107 100)(111 114 133 136)(112 135 134 113)(115 120 137 132)(116 131 138 119)(117 118 139 140)(121 146 149 128)(122 127 150 145)(123 144 141 126)(124 125 142 143)(129 148 147 130)```

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 4R 5A 5B 10A ··· 10F 10G ··· 10N 20A ··· 20L order 1 2 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 4 4 4 4 10 10 2 ··· 2 5 5 5 5 10 10 10 10 20 20 20 20 2 2 2 ··· 2 8 ··· 8 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D4×D5 D4⋊2D5 kernel C42.238D10 C20⋊2Q8 D5×C42 D4×Dic5 C20.17D4 C20⋊2D4 C5×C4⋊1D4 C2×D4⋊2D5 C4×D5 C4⋊1D4 C20 C42 C2×D4 C4 C4 # reps 1 1 1 4 2 4 1 2 4 2 8 2 12 4 8

Matrix representation of C42.238D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 0 0 0 40 9
,
 32 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 0 0 0 40 9
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 7 7 0 0 0 0 34 40 0 0 0 0 0 0 40 18 0 0 0 0 0 1
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 7 7 0 0 0 0 40 34 0 0 0 0 0 0 1 23 0 0 0 0 32 40

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,40,0,0,0,0,0,9],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,40,0,0,0,0,0,9],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,40,0,0,0,0,0,18,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,1,32,0,0,0,0,23,40] >;`

C42.238D10 in GAP, Magma, Sage, TeX

`C_4^2._{238}D_{10}`
`% in TeX`

`G:=Group("C4^2.238D10");`
`// GroupNames label`

`G:=SmallGroup(320,1388);`
`// by ID`

`G=gap.SmallGroup(320,1388);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,675,570,185,12550]);`
`// Polycyclic`

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

׿
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