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## G = C42.20D10order 320 = 26·5

### 20th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C42.20D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C4.D20 — C42.20D10
 Lower central C5 — C10 — C2×C20 — C42.20D10
 Upper central C1 — C22 — C42 — C8⋊C4

Generators and relations for C42.20D10
G = < a,b,c,d | a4=b4=1, c10=a2b-1, d2=a2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=bc9 >

Subgroups: 494 in 100 conjugacy classes, 39 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C8⋊C4, D4⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C40, Dic10, D20, C2×Dic5, C2×C20, C22×D5, C42.28C22, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×Dic10, C2×Dic10, C2×D20, C20.44D4, D205C4, C5×C8⋊C4, C202Q8, C4.D20, C42.20D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C8⋊C22, C8.C22, D20, C22×D5, C42.28C22, C2×D20, C4○D20, C4.D20, C8⋊D10, C8.D10, C42.20D10

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

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A ··· 20H 20I ··· 20P 40A ··· 40P order 1 2 2 2 2 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 40 2 2 4 4 40 40 40 2 2 4 4 4 4 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D20 C4○D20 C8⋊C22 C8.C22 C8⋊D10 C8.D10 kernel C42.20D10 C20.44D4 D20⋊5C4 C5×C8⋊C4 C20⋊2Q8 C4.D20 C2×C20 C8⋊C4 C20 C42 C2×C8 C2×C4 C4 C10 C10 C2 C2 # reps 1 2 2 1 1 1 2 2 4 2 4 8 16 1 1 4 4

Matrix representation of C42.20D10 in GL6(𝔽41)

 40 21 0 0 0 0 37 1 0 0 0 0 0 0 27 13 18 19 0 0 28 23 15 40 0 0 22 2 5 28 0 0 8 20 22 27
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 2 13 0 0 0 0 28 39 0 0 0 0 0 0 11 13 0 0 0 0 19 30
,
 9 16 0 0 0 0 36 32 0 0 0 0 0 0 13 13 0 18 0 0 28 9 20 23 0 0 21 14 19 28 0 0 3 7 22 0
,
 32 25 0 0 0 0 0 9 0 0 0 0 0 0 21 20 14 7 0 0 23 20 1 35 0 0 15 38 3 23 0 0 23 6 37 38

`G:=sub<GL(6,GF(41))| [40,37,0,0,0,0,21,1,0,0,0,0,0,0,27,28,22,8,0,0,13,23,2,20,0,0,18,15,5,22,0,0,19,40,28,27],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,11,19,0,0,0,0,13,30],[9,36,0,0,0,0,16,32,0,0,0,0,0,0,13,28,21,3,0,0,13,9,14,7,0,0,0,20,19,22,0,0,18,23,28,0],[32,0,0,0,0,0,25,9,0,0,0,0,0,0,21,23,15,23,0,0,20,20,38,6,0,0,14,1,3,37,0,0,7,35,23,38] >;`

C42.20D10 in GAP, Magma, Sage, TeX

`C_4^2._{20}D_{10}`
`% in TeX`

`G:=Group("C4^2.20D10");`
`// GroupNames label`

`G:=SmallGroup(320,341);`
`// by ID`

`G=gap.SmallGroup(320,341);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,387,142,1123,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2*b^-1,d^2=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b*c^9>;`
`// generators/relations`

׿
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