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## G = C42.87D10order 320 = 26·5

### 87th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C42.87D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C22×Dic10 — C42.87D10
 Lower central C5 — C10 — C42.87D10
 Upper central C1 — C22 — C42⋊C2

Generators and relations for C42.87D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=c-1 >

Subgroups: 686 in 266 conjugacy classes, 151 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×16], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], Q8 [×16], C23, C10, C10 [×2], C10 [×2], C42 [×2], C42 [×10], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×10], C22×C4, C22×C4 [×2], C2×Q8 [×12], Dic5 [×8], Dic5 [×4], C20 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2, C42⋊C2 [×5], C4×Q8 [×8], C22×Q8, Dic10 [×16], C2×Dic5 [×16], C2×C20 [×2], C2×C20 [×8], C22×C10, C23.32C23, C4×Dic5 [×10], C10.D4 [×8], C4⋊Dic5 [×2], C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10 [×12], C22×Dic5 [×2], C22×C20, C4×Dic10 [×4], C23.11D10 [×4], Dic53Q8 [×4], C23.21D10, C5×C42⋊C2, C22×Dic10, C42.87D10
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, 2- 1+4 [×2], C4×D5 [×4], C22×D5 [×7], C23.32C23, C2×C4×D5 [×6], C23×D5, D5×C22×C4, D4.10D10 [×2], C42.87D10

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

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4L 4M ··· 4AB 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20AB order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 2 ··· 2 10 ··· 10 2 2 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 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 C4 D5 D10 D10 D10 D10 C4×D5 2- 1+4 D4.10D10 kernel C42.87D10 C4×Dic10 C23.11D10 Dic5⋊3Q8 C23.21D10 C5×C42⋊C2 C22×Dic10 C2×Dic10 C42⋊C2 C42 C22⋊C4 C4⋊C4 C22×C4 C2×C4 C10 C2 # reps 1 4 4 4 1 1 1 16 2 4 4 4 2 16 2 8

Matrix representation of C42.87D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 2 32 0 0 0 0 37 39 0 0 0 0 0 0 11 32 0 0 0 0 9 30
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 23 23 0 0 0 9 36 38 0 0 0 0 32 0 0 0 0 0 0 32
,
 0 6 0 0 0 0 34 7 0 0 0 0 0 0 6 7 0 0 0 0 35 0 0 0 0 0 19 10 1 34 0 0 28 38 7 34
,
 26 38 0 0 0 0 20 15 0 0 0 0 0 0 13 22 0 0 0 0 37 28 0 0 0 0 6 14 22 19 0 0 7 8 9 19

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,2,37,0,0,0,0,32,39,0,0,0,0,0,0,11,9,0,0,0,0,32,30],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,23,36,32,0,0,0,23,38,0,32],[0,34,0,0,0,0,6,7,0,0,0,0,0,0,6,35,19,28,0,0,7,0,10,38,0,0,0,0,1,7,0,0,0,0,34,34],[26,20,0,0,0,0,38,15,0,0,0,0,0,0,13,37,6,7,0,0,22,28,14,8,0,0,0,0,22,9,0,0,0,0,19,19] >;`

C42.87D10 in GAP, Magma, Sage, TeX

`C_4^2._{87}D_{10}`
`% in TeX`

`G:=Group("C4^2.87D10");`
`// GroupNames label`

`G:=SmallGroup(320,1188);`
`// by ID`

`G=gap.SmallGroup(320,1188);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,184,570,80,12550]);`
`// Polycyclic`

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

׿
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