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## G = C42.7F5order 320 = 26·5

### 4th non-split extension by C42 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.7F5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — Dic5⋊C8 — C42.7F5
 Lower central C5 — C2×C10 — C42.7F5
 Upper central C1 — C22 — C42

Generators and relations for C42.7F5
G = < a,b,c,d | a4=b4=c5=1, d4=a2b2, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b-1, dcd-1=c3 >

Subgroups: 378 in 110 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C42, C42, C2×C8, C22×C4, Dic5, Dic5, C20, D10, D10, C2×C10, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C42.6C4, C4×Dic5, C4×C20, C2×C5⋊C8, C2×C4×D5, C10.C42, D10⋊C8, Dic5⋊C8, D5×C42, C42.7F5
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C4○D4, F5, C42⋊C2, C2×M4(2), C2×F5, C42.6C4, C22×F5, D5⋊M4(2), D10.C23, C42.7F5

Smallest permutation representation of C42.7F5
On 160 points
Generators in S160
```(1 50 61 100)(2 97 62 55)(3 52 63 102)(4 99 64 49)(5 54 57 104)(6 101 58 51)(7 56 59 98)(8 103 60 53)(9 108 76 65)(10 70 77 105)(11 110 78 67)(12 72 79 107)(13 112 80 69)(14 66 73 109)(15 106 74 71)(16 68 75 111)(17 140 115 44)(18 41 116 137)(19 142 117 46)(20 43 118 139)(21 144 119 48)(22 45 120 141)(23 138 113 42)(24 47 114 143)(25 40 121 157)(26 154 122 37)(27 34 123 159)(28 156 124 39)(29 36 125 153)(30 158 126 33)(31 38 127 155)(32 160 128 35)(81 149 90 134)(82 131 91 146)(83 151 92 136)(84 133 93 148)(85 145 94 130)(86 135 95 150)(87 147 96 132)(88 129 89 152)
(1 98 57 52)(2 103 58 49)(3 100 59 54)(4 97 60 51)(5 102 61 56)(6 99 62 53)(7 104 63 50)(8 101 64 55)(9 30 80 122)(10 27 73 127)(11 32 74 124)(12 29 75 121)(13 26 76 126)(14 31 77 123)(15 28 78 128)(16 25 79 125)(17 135 119 146)(18 132 120 151)(19 129 113 148)(20 134 114 145)(21 131 115 150)(22 136 116 147)(23 133 117 152)(24 130 118 149)(33 112 154 65)(34 109 155 70)(35 106 156 67)(36 111 157 72)(37 108 158 69)(38 105 159 66)(39 110 160 71)(40 107 153 68)(41 87 141 92)(42 84 142 89)(43 81 143 94)(44 86 144 91)(45 83 137 96)(46 88 138 93)(47 85 139 90)(48 82 140 95)
(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)| (1,50,61,100)(2,97,62,55)(3,52,63,102)(4,99,64,49)(5,54,57,104)(6,101,58,51)(7,56,59,98)(8,103,60,53)(9,108,76,65)(10,70,77,105)(11,110,78,67)(12,72,79,107)(13,112,80,69)(14,66,73,109)(15,106,74,71)(16,68,75,111)(17,140,115,44)(18,41,116,137)(19,142,117,46)(20,43,118,139)(21,144,119,48)(22,45,120,141)(23,138,113,42)(24,47,114,143)(25,40,121,157)(26,154,122,37)(27,34,123,159)(28,156,124,39)(29,36,125,153)(30,158,126,33)(31,38,127,155)(32,160,128,35)(81,149,90,134)(82,131,91,146)(83,151,92,136)(84,133,93,148)(85,145,94,130)(86,135,95,150)(87,147,96,132)(88,129,89,152), (1,98,57,52)(2,103,58,49)(3,100,59,54)(4,97,60,51)(5,102,61,56)(6,99,62,53)(7,104,63,50)(8,101,64,55)(9,30,80,122)(10,27,73,127)(11,32,74,124)(12,29,75,121)(13,26,76,126)(14,31,77,123)(15,28,78,128)(16,25,79,125)(17,135,119,146)(18,132,120,151)(19,129,113,148)(20,134,114,145)(21,131,115,150)(22,136,116,147)(23,133,117,152)(24,130,118,149)(33,112,154,65)(34,109,155,70)(35,106,156,67)(36,111,157,72)(37,108,158,69)(38,105,159,66)(39,110,160,71)(40,107,153,68)(41,87,141,92)(42,84,142,89)(43,81,143,94)(44,86,144,91)(45,83,137,96)(46,88,138,93)(47,85,139,90)(48,82,140,95), (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( (1,50,61,100)(2,97,62,55)(3,52,63,102)(4,99,64,49)(5,54,57,104)(6,101,58,51)(7,56,59,98)(8,103,60,53)(9,108,76,65)(10,70,77,105)(11,110,78,67)(12,72,79,107)(13,112,80,69)(14,66,73,109)(15,106,74,71)(16,68,75,111)(17,140,115,44)(18,41,116,137)(19,142,117,46)(20,43,118,139)(21,144,119,48)(22,45,120,141)(23,138,113,42)(24,47,114,143)(25,40,121,157)(26,154,122,37)(27,34,123,159)(28,156,124,39)(29,36,125,153)(30,158,126,33)(31,38,127,155)(32,160,128,35)(81,149,90,134)(82,131,91,146)(83,151,92,136)(84,133,93,148)(85,145,94,130)(86,135,95,150)(87,147,96,132)(88,129,89,152), (1,98,57,52)(2,103,58,49)(3,100,59,54)(4,97,60,51)(5,102,61,56)(6,99,62,53)(7,104,63,50)(8,101,64,55)(9,30,80,122)(10,27,73,127)(11,32,74,124)(12,29,75,121)(13,26,76,126)(14,31,77,123)(15,28,78,128)(16,25,79,125)(17,135,119,146)(18,132,120,151)(19,129,113,148)(20,134,114,145)(21,131,115,150)(22,136,116,147)(23,133,117,152)(24,130,118,149)(33,112,154,65)(34,109,155,70)(35,106,156,67)(36,111,157,72)(37,108,158,69)(38,105,159,66)(39,110,160,71)(40,107,153,68)(41,87,141,92)(42,84,142,89)(43,81,143,94)(44,86,144,91)(45,83,137,96)(46,88,138,93)(47,85,139,90)(48,82,140,95), (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([[(1,50,61,100),(2,97,62,55),(3,52,63,102),(4,99,64,49),(5,54,57,104),(6,101,58,51),(7,56,59,98),(8,103,60,53),(9,108,76,65),(10,70,77,105),(11,110,78,67),(12,72,79,107),(13,112,80,69),(14,66,73,109),(15,106,74,71),(16,68,75,111),(17,140,115,44),(18,41,116,137),(19,142,117,46),(20,43,118,139),(21,144,119,48),(22,45,120,141),(23,138,113,42),(24,47,114,143),(25,40,121,157),(26,154,122,37),(27,34,123,159),(28,156,124,39),(29,36,125,153),(30,158,126,33),(31,38,127,155),(32,160,128,35),(81,149,90,134),(82,131,91,146),(83,151,92,136),(84,133,93,148),(85,145,94,130),(86,135,95,150),(87,147,96,132),(88,129,89,152)], [(1,98,57,52),(2,103,58,49),(3,100,59,54),(4,97,60,51),(5,102,61,56),(6,99,62,53),(7,104,63,50),(8,101,64,55),(9,30,80,122),(10,27,73,127),(11,32,74,124),(12,29,75,121),(13,26,76,126),(14,31,77,123),(15,28,78,128),(16,25,79,125),(17,135,119,146),(18,132,120,151),(19,129,113,148),(20,134,114,145),(21,131,115,150),(22,136,116,147),(23,133,117,152),(24,130,118,149),(33,112,154,65),(34,109,155,70),(35,106,156,67),(36,111,157,72),(37,108,158,69),(38,105,159,66),(39,110,160,71),(40,107,153,68),(41,87,141,92),(42,84,142,89),(43,81,143,94),(44,86,144,91),(45,83,137,96),(46,88,138,93),(47,85,139,90),(48,82,140,95)], [(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 1 2 2 2 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 M4(2) C4○D4 M4(2) F5 C2×F5 D5⋊M4(2) D10.C23 kernel C42.7F5 C10.C42 D10⋊C8 Dic5⋊C8 D5×C42 C4×Dic5 C4×C20 C2×C4×D5 Dic5 Dic5 D10 C42 C2×C4 C2 C2 # reps 1 2 2 2 1 2 2 4 4 4 4 1 3 8 4

Matrix representation of C42.7F5 in GL6(𝔽41)

 1 0 0 0 0 0 23 40 0 0 0 0 0 0 34 14 0 27 0 0 0 7 14 27 0 0 27 14 7 0 0 0 27 0 14 34
,
 32 0 0 0 0 0 39 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 0 0 9
,
 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
,
 8 10 0 0 0 0 5 33 0 0 0 0 0 0 24 20 24 7 0 0 7 27 31 31 0 0 14 10 10 14 0 0 34 34 17 21

`G:=sub<GL(6,GF(41))| [1,23,0,0,0,0,0,40,0,0,0,0,0,0,34,0,27,27,0,0,14,7,14,0,0,0,0,14,7,14,0,0,27,27,0,34],[32,39,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[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],[8,5,0,0,0,0,10,33,0,0,0,0,0,0,24,7,14,34,0,0,20,27,10,34,0,0,24,31,10,17,0,0,7,31,14,21] >;`

C42.7F5 in GAP, Magma, Sage, TeX

`C_4^2._7F_5`
`% in TeX`

`G:=Group("C4^2.7F5");`
`// GroupNames label`

`G:=SmallGroup(320,1022);`
`// by ID`

`G=gap.SmallGroup(320,1022);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,344,758,100,136,6278,1595]);`
`// Polycyclic`

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

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