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## G = C42.14F5order 320 = 26·5

### 11st non-split extension by C42 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.14F5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C2×C4.F5 — C42.14F5
 Lower central C5 — C2×C10 — C42.14F5
 Upper central C1 — C22 — C42

Generators and relations for C42.14F5
G = < a,b,c,d | a4=b4=c5=1, d4=a2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=c3 >

Subgroups: 426 in 126 conjugacy classes, 56 normal (18 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, M4(2), C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4⋊C8, C2×C42, C2×M4(2), C5⋊C8, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4⋊M4(2), C4×Dic5, C4×Dic5, C4×C20, C4.F5, C2×C5⋊C8, C2×C4×D5, C2×C4×D5, Dic5⋊C8, D5×C42, C2×C4.F5, C42.14F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C2×M4(2), C2×F5, C4⋊M4(2), C4⋊F5, C22×F5, D5⋊M4(2), C2×C4⋊F5, C42.14F5

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 4 4 4 4 type + + + + + - + + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 M4(2) F5 C2×F5 C4⋊F5 D5⋊M4(2) kernel C42.14F5 Dic5⋊C8 D5×C42 C2×C4.F5 C4×Dic5 C4×C20 C2×C4×D5 C4×D5 C4×D5 Dic5 C42 C2×C4 C4 C2 # reps 1 4 1 2 2 2 4 2 2 8 1 3 4 8

Matrix representation of C42.14F5 in GL6(𝔽41)

 32 0 0 0 0 0 23 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 2 40 0 0 0 0 0 0 7 0 14 14 0 0 27 34 27 0 0 0 0 27 34 27 0 0 14 14 0 7
,
 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
,
 10 31 0 0 0 0 5 31 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 40 0 0 0 0 1 1 1 1

`G:=sub<GL(6,GF(41))| [32,23,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,40,0,0,0,0,0,0,7,27,0,14,0,0,0,34,27,14,0,0,14,27,34,0,0,0,14,0,27,7],[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],[10,5,0,0,0,0,31,31,0,0,0,0,0,0,40,0,0,1,0,0,0,0,40,1,0,0,0,0,0,1,0,0,0,40,0,1] >;`

C42.14F5 in GAP, Magma, Sage, TeX

`C_4^2._{14}F_5`
`% in TeX`

`G:=Group("C4^2.14F5");`
`// GroupNames label`

`G:=SmallGroup(320,1020);`
`// by ID`

`G=gap.SmallGroup(320,1020);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,232,422,268,136,6278,1595]);`
`// 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,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^3>;`
`// generators/relations`

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