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## G = C42.237D10order 320 = 26·5

### 57th non-split extension by C42 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.237D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C42 — C42.237D10
 Lower central C5 — C2×C10 — C42.237D10
 Upper central C1 — C22 — C42.C2

Generators and relations for C42.237D10
G = < a,b,c,d | a4=b4=d2=1, c10=a2, ab=ba, cac-1=dad=ab2, cbc-1=dbd=a2b, dcd=a2c9 >

Subgroups: 830 in 234 conjugacy classes, 99 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×15], D4 [×6], Q8 [×2], C23 [×3], D5 [×4], C10 [×3], C42, C42 [×5], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×5], C2×D4 [×3], C2×Q8, Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×6], D10 [×2], D10 [×8], C2×C10, C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C422C2 [×2], Dic10 [×2], C4×D5 [×10], D20 [×6], C2×Dic5 [×3], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×4], C22×D5, C22×D5 [×2], C23.36C23, C4×Dic5 [×3], C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×10], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×4], C2×Dic10, C2×C4×D5 [×3], C2×C4×D5 [×2], C2×D20, C2×D20 [×2], D5×C42, C4.D20, Dic53Q8, C4⋊C47D5 [×2], D208C4, D208C4 [×2], D10.13D4 [×2], C4⋊D20, D102Q8, C4⋊C4⋊D5 [×2], C5×C42.C2, C42.237D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C2×C4○D4 [×3], C22×D5 [×7], C23.36C23, Q82D5 [×2], C23×D5, C2×Q82D5, D5×C4○D4 [×2], C42.237D10

Smallest permutation representation of C42.237D10
On 160 points
Generators in S160
```(1 73 11 63)(2 39 12 29)(3 75 13 65)(4 21 14 31)(5 77 15 67)(6 23 16 33)(7 79 17 69)(8 25 18 35)(9 61 19 71)(10 27 20 37)(22 47 32 57)(24 49 34 59)(26 51 36 41)(28 53 38 43)(30 55 40 45)(42 62 52 72)(44 64 54 74)(46 66 56 76)(48 68 58 78)(50 70 60 80)(81 159 91 149)(82 119 92 109)(83 141 93 151)(84 101 94 111)(85 143 95 153)(86 103 96 113)(87 145 97 155)(88 105 98 115)(89 147 99 157)(90 107 100 117)(102 140 112 130)(104 122 114 132)(106 124 116 134)(108 126 118 136)(110 128 120 138)(121 154 131 144)(123 156 133 146)(125 158 135 148)(127 160 137 150)(129 142 139 152)
(1 155 53 114)(2 146 54 105)(3 157 55 116)(4 148 56 107)(5 159 57 118)(6 150 58 109)(7 141 59 120)(8 152 60 111)(9 143 41 102)(10 154 42 113)(11 145 43 104)(12 156 44 115)(13 147 45 106)(14 158 46 117)(15 149 47 108)(16 160 48 119)(17 151 49 110)(18 142 50 101)(19 153 51 112)(20 144 52 103)(21 125 76 100)(22 136 77 91)(23 127 78 82)(24 138 79 93)(25 129 80 84)(26 140 61 95)(27 131 62 86)(28 122 63 97)(29 133 64 88)(30 124 65 99)(31 135 66 90)(32 126 67 81)(33 137 68 92)(34 128 69 83)(35 139 70 94)(36 130 71 85)(37 121 72 96)(38 132 73 87)(39 123 74 98)(40 134 75 89)
(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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 74)(22 73)(23 72)(24 71)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 80)(36 79)(37 78)(38 77)(39 76)(40 75)(41 49)(42 48)(43 47)(44 46)(50 60)(51 59)(52 58)(53 57)(54 56)(81 132)(82 131)(83 130)(84 129)(85 128)(86 127)(87 126)(88 125)(89 124)(90 123)(91 122)(92 121)(93 140)(94 139)(95 138)(96 137)(97 136)(98 135)(99 134)(100 133)(102 120)(103 119)(104 118)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(141 143)(144 160)(145 159)(146 158)(147 157)(148 156)(149 155)(150 154)(151 153)```

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 C4○D4 D10 D10 Q8⋊2D5 D5×C4○D4 kernel C42.237D10 D5×C42 C4.D20 Dic5⋊3Q8 C4⋊C4⋊7D5 D20⋊8C4 D10.13D4 C4⋊D20 D10⋊2Q8 C4⋊C4⋊D5 C5×C42.C2 C42.C2 Dic5 C20 D10 C42 C4⋊C4 C4 C2 # reps 1 1 1 1 2 3 2 1 1 2 1 2 4 4 4 2 12 4 8

Matrix representation of C42.237D10 in GL6(𝔽41)

 32 0 0 0 0 0 0 32 0 0 0 0 0 0 40 33 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 39 0 0 0 0 0 40 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 39 0 0 0 0 1 40 0 0 0 0 0 0 1 0 0 0 0 0 10 40 0 0 0 0 0 0 1 6 0 0 0 0 35 6
,
 1 0 0 0 0 0 1 40 0 0 0 0 0 0 1 0 0 0 0 0 10 40 0 0 0 0 0 0 40 0 0 0 0 0 6 1

`G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,33,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,1,10,0,0,0,0,0,40,0,0,0,0,0,0,1,35,0,0,0,0,6,6],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,10,0,0,0,0,0,40,0,0,0,0,0,0,40,6,0,0,0,0,0,1] >;`

C42.237D10 in GAP, Magma, Sage, TeX

`C_4^2._{237}D_{10}`
`% in TeX`

`G:=Group("C4^2.237D10");`
`// GroupNames label`

`G:=SmallGroup(320,1363);`
`// by ID`

`G=gap.SmallGroup(320,1363);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,1123,346,297,136,12550]);`
`// Polycyclic`

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

׿
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