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## G = C42.5F5order 320 = 26·5

### 2nd non-split extension by C42 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C42.5F5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C2×D5⋊C8 — C42.5F5
 Lower central C5 — C10 — C42.5F5
 Upper central C1 — C2×C4 — C42

Generators and relations for C42.5F5
G = < a,b,c,d | a4=b4=c5=1, d4=a2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 426 in 146 conjugacy classes, 76 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, D5, C10, C10, C42, C42, C2×C8, C22×C4, Dic5, Dic5, C20, C20, D10, C2×C10, C8⋊C4, C2×C42, C22×C8, C5⋊C8, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C2×C8⋊C4, C4×Dic5, C4×Dic5, C4×C20, D5⋊C8, C2×C5⋊C8, C2×C4×D5, C2×C4×D5, C10.C42, D5×C42, C2×D5⋊C8, C42.5F5
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, F5, C8⋊C4, C2×C42, C2×M4(2), C2×F5, C2×C8⋊C4, C4×F5, C22×F5, D5⋊M4(2), C2×C4×F5, C42.5F5

Smallest permutation representation of C42.5F5
On 160 points
Generators in S160
```(1 134 5 130)(2 135 6 131)(3 136 7 132)(4 129 8 133)(9 158 13 154)(10 159 14 155)(11 160 15 156)(12 153 16 157)(17 96 21 92)(18 89 22 93)(19 90 23 94)(20 91 24 95)(25 149 29 145)(26 150 30 146)(27 151 31 147)(28 152 32 148)(33 81 37 85)(34 82 38 86)(35 83 39 87)(36 84 40 88)(41 57 45 61)(42 58 46 62)(43 59 47 63)(44 60 48 64)(49 99 53 103)(50 100 54 104)(51 101 55 97)(52 102 56 98)(65 137 69 141)(66 138 70 142)(67 139 71 143)(68 140 72 144)(73 113 77 117)(74 114 78 118)(75 115 79 119)(76 116 80 120)(105 123 109 127)(106 124 110 128)(107 125 111 121)(108 126 112 122)
(1 18 132 95)(2 23 133 92)(3 20 134 89)(4 17 135 94)(5 22 136 91)(6 19 129 96)(7 24 130 93)(8 21 131 90)(9 97 80 122)(10 102 73 127)(11 99 74 124)(12 104 75 121)(13 101 76 126)(14 98 77 123)(15 103 78 128)(16 100 79 125)(25 139 33 47)(26 144 34 44)(27 141 35 41)(28 138 36 46)(29 143 37 43)(30 140 38 48)(31 137 39 45)(32 142 40 42)(49 118 106 156)(50 115 107 153)(51 120 108 158)(52 117 109 155)(53 114 110 160)(54 119 111 157)(55 116 112 154)(56 113 105 159)(57 151 65 83)(58 148 66 88)(59 145 67 85)(60 150 68 82)(61 147 69 87)(62 152 70 84)(63 149 71 81)(64 146 72 86)
(1 12 57 143 113)(2 144 13 114 58)(3 115 137 59 14)(4 60 116 15 138)(5 16 61 139 117)(6 140 9 118 62)(7 119 141 63 10)(8 64 120 11 142)(17 150 112 103 36)(18 104 151 37 105)(19 38 97 106 152)(20 107 39 145 98)(21 146 108 99 40)(22 100 147 33 109)(23 34 101 110 148)(24 111 35 149 102)(25 52 91 125 87)(26 126 53 88 92)(27 81 127 93 54)(28 94 82 55 128)(29 56 95 121 83)(30 122 49 84 96)(31 85 123 89 50)(32 90 86 51 124)(41 71 73 130 157)(42 131 72 158 74)(43 159 132 75 65)(44 76 160 66 133)(45 67 77 134 153)(46 135 68 154 78)(47 155 136 79 69)(48 80 156 70 129)
(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)```

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 4 4 4 4 type + + + + + + image C1 C2 C2 C2 C4 C4 C4 C4 M4(2) F5 C2×F5 C4×F5 D5⋊M4(2) kernel C42.5F5 C10.C42 D5×C42 C2×D5⋊C8 C4×Dic5 C4×C20 D5⋊C8 C2×C4×D5 D10 C42 C2×C4 C4 C2 # reps 1 4 1 2 2 2 16 4 8 1 3 4 8

Matrix representation of C42.5F5 in GL6(𝔽41)

 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32
,
 18 2 0 0 0 0 22 23 0 0 0 0 0 0 19 0 38 38 0 0 3 22 3 0 0 0 0 3 22 3 0 0 38 38 0 19
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 3 12 0 0 0 0 19 38 0 0 0 0 0 0 17 0 26 26 0 0 26 26 0 17 0 0 15 32 15 0 0 0 24 9 9 24

`G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[18,22,0,0,0,0,2,23,0,0,0,0,0,0,19,3,0,38,0,0,0,22,3,38,0,0,38,3,22,0,0,0,38,0,3,19],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[3,19,0,0,0,0,12,38,0,0,0,0,0,0,17,26,15,24,0,0,0,26,32,9,0,0,26,0,15,9,0,0,26,17,0,24] >;`

C42.5F5 in GAP, Magma, Sage, TeX

`C_4^2._5F_5`
`% in TeX`

`G:=Group("C4^2.5F5");`
`// GroupNames label`

`G:=SmallGroup(320,1014);`
`// by ID`

`G=gap.SmallGroup(320,1014);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,100,136,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^5=1,d^4=a^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;`
`// generators/relations`

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