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## G = C42.90D10order 320 = 26·5

### 90th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.90D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C4⋊Dic5 — C42.90D10
 Lower central C5 — C2×C10 — C42.90D10
 Upper central C1 — C22 — C42⋊C2

Generators and relations for C42.90D10
G = < a,b,c,d | a4=b4=c10=1, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 638 in 206 conjugacy classes, 111 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×10], Q8 [×4], C23, C10 [×3], C10 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×18], C22×C4, C22×C4 [×2], C2×Q8 [×4], Dic5 [×8], C20 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4, C42⋊C2, C42⋊C2, C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8 [×4], Dic10 [×4], C2×Dic5 [×8], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×8], C22×C10, C23.41C23, C4×Dic5 [×2], C10.D4 [×8], C4⋊Dic5 [×2], C4⋊Dic5 [×8], C23.D5 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10 [×4], C22×Dic5 [×2], C22×C20, C202Q8 [×2], C20.6Q8 [×2], Dic5.14D4 [×4], C20⋊Q8 [×2], C4.Dic10 [×2], C2×C4⋊Dic5, C23.21D10, C5×C42⋊C2, C42.90D10
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C24, D10 [×7], C22×Q8, 2+ 1+4, 2- 1+4, Dic10 [×4], C22×D5 [×7], C23.41C23, C2×Dic10 [×6], C23×D5, C22×Dic10, D48D10, D4.10D10, C42.90D10

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + + + + - + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 Q8 D5 D10 D10 D10 D10 Dic10 2+ 1+4 2- 1+4 D4⋊8D10 D4.10D10 kernel C42.90D10 C20⋊2Q8 C20.6Q8 Dic5.14D4 C20⋊Q8 C4.Dic10 C2×C4⋊Dic5 C23.21D10 C5×C42⋊C2 C2×C20 C42⋊C2 C42 C22⋊C4 C4⋊C4 C22×C4 C2×C4 C10 C10 C2 C2 # reps 1 2 2 4 2 2 1 1 1 4 2 4 4 4 2 16 1 1 4 4

Matrix representation of C42.90D10 in GL6(𝔽41)

 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 40 0 0 0 0 0 0 40 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 13 0 0 0 0 28 39 0 0 0 0 0 0 2 13 0 0 0 0 28 39
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 35 35 0 0 0 0 6 40 0 0 0 0 0 0 6 6 0 0 0 0 35 1
,
 0 9 0 0 0 0 9 0 0 0 0 0 0 0 37 14 0 0 0 0 31 4 0 0 0 0 0 0 37 14 0 0 0 0 31 4

`G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,40,0,0,0,0,0,0,40,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,2,28,0,0,0,0,13,39],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,6,35,0,0,0,0,6,1],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,37,31,0,0,0,0,14,4,0,0,0,0,0,0,37,31,0,0,0,0,14,4] >;`

C42.90D10 in GAP, Magma, Sage, TeX

`C_4^2._{90}D_{10}`
`% in TeX`

`G:=Group("C4^2.90D10");`
`// GroupNames label`

`G:=SmallGroup(320,1191);`
`// by ID`

`G=gap.SmallGroup(320,1191);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,758,184,675,570,80,12550]);`
`// Polycyclic`

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

׿
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