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## G = C42.12F5order 320 = 26·5

### 9th non-split extension by C42 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C42.12F5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C20⋊C8 — C42.12F5
 Lower central C5 — C10 — C42.12F5
 Upper central C1 — C2×C4 — C42

Generators and relations for C42.12F5
G = < a,b,c,d | a4=b4=c5=1, d4=a2b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 378 in 118 conjugacy classes, 58 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, 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, C20⋊C8, D10⋊C8, D5×C42, C42.12F5
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, C4.F5, C22×F5, C2×D5⋊C8, C2×C4.F5, D10.C23, C42.12F5

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

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

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

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

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 C4○D4 M4(2) F5 C2×F5 D5⋊C8 C4.F5 D10.C23 kernel C42.12F5 C4×C5⋊C8 C20⋊C8 D10⋊C8 D5×C42 C4×Dic5 C4×C20 C2×C4×D5 C4×D5 Dic5 C20 C42 C2×C4 C4 C4 C2 # reps 1 2 2 2 1 2 2 4 16 4 4 1 3 4 4 4

Matrix representation of C42.12F5 in GL6(𝔽41)

 32 0 0 0 0 0 18 9 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
,
 40 0 0 0 0 0 0 40 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 40 40 40 40 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 5 5 0 0 0 0 26 36 0 0 0 0 0 0 6 12 36 17 0 0 24 5 29 35 0 0 24 30 36 19 0 0 6 30 11 35

`G:=sub<GL(6,GF(41))| [32,18,0,0,0,0,0,9,0,0,0,0,0,0,34,14,0,27,0,0,0,7,14,27,0,0,27,14,7,0,0,0,27,0,14,34],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[5,26,0,0,0,0,5,36,0,0,0,0,0,0,6,24,24,6,0,0,12,5,30,30,0,0,36,29,36,11,0,0,17,35,19,35] >;`

C42.12F5 in GAP, Magma, Sage, TeX

`C_4^2._{12}F_5`
`% in TeX`

`G:=Group("C4^2.12F5");`
`// GroupNames label`

`G:=SmallGroup(320,1018);`
`// by ID`

`G=gap.SmallGroup(320,1018);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,120,268,136,6278,1595]);`
`// 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^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;`
`// generators/relations`

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