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## G = C42.6F5order 320 = 26·5

### 3rd non-split extension by C42 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C42.6F5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C4×C5⋊C8 — C42.6F5
 Lower central C5 — C10 — C42.6F5
 Upper central C1 — C2×C4 — C42

Generators and relations for C42.6F5
G = < a,b,c,d | a4=b4=c5=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 378 in 118 conjugacy classes, 56 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C42, C42, C2×C8, C22×C4, Dic5, C20, C20, D10, D10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C5⋊C8, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, C42.12C4, C4×Dic5, C4×C20, C2×C5⋊C8, C2×C4×D5, C4×C5⋊C8, D10⋊C8, Dic5⋊C8, D5×C42, C42.6F5
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, M4(2), C22×C4, C4○D4, F5, C42⋊C2, C22×C8, C2×M4(2), C2×F5, C42.12C4, D5⋊C8, C22×F5, C2×D5⋊C8, D5⋊M4(2), D10.C23, C42.6F5

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 M4(2) C4○D4 F5 C2×F5 D5⋊C8 D5⋊M4(2) D10.C23 kernel C42.6F5 C4×C5⋊C8 D10⋊C8 Dic5⋊C8 D5×C42 C4×Dic5 C4×C20 C2×C4×D5 C4×D5 Dic5 Dic5 C42 C2×C4 C4 C2 C2 # reps 1 2 2 2 1 2 2 4 16 4 4 1 3 4 4 4

Matrix representation of C42.6F5 in GL6(𝔽41)

 9 0 0 0 0 0 0 9 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
,
 32 0 0 0 0 0 0 9 0 0 0 0 0 0 22 38 0 3 0 0 0 19 38 3 0 0 3 38 19 0 0 0 3 0 38 22
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 1 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40
,
 0 1 0 0 0 0 9 0 0 0 0 0 0 0 4 4 24 26 0 0 28 30 7 30 0 0 11 34 11 13 0 0 15 17 37 37

`G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,22,0,3,3,0,0,38,19,38,0,0,0,0,38,19,38,0,0,3,3,0,22],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[0,9,0,0,0,0,1,0,0,0,0,0,0,0,4,28,11,15,0,0,4,30,34,17,0,0,24,7,11,37,0,0,26,30,13,37] >;`

C42.6F5 in GAP, Magma, Sage, TeX

`C_4^2._6F_5`
`% in TeX`

`G:=Group("C4^2.6F5");`
`// GroupNames label`

`G:=SmallGroup(320,1016);`
`// by ID`

`G=gap.SmallGroup(320,1016);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,422,184,136,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^5=1,d^4=b^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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