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## G = C42.126D10order 320 = 26·5

### 126th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C42.126D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — C42.126D10
 Lower central C5 — C10 — C42.126D10
 Upper central C1 — C22 — C4×Q8

Generators and relations for C42.126D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c9 >

Subgroups: 958 in 294 conjugacy classes, 151 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C2×C4○D4, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C23.33C23, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, D10⋊C4, C4×C20, C5×C4⋊C4, C2×C4×D5, C2×D20, Q82D5, Q8×C10, C42⋊D5, C4×D20, D5×C4⋊C4, D208C4, Q8×Dic5, Q8×C20, C2×Q82D5, C42.126D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C23×C4, 2+ 1+4, 2- 1+4, C4×D5, C22×D5, C23.33C23, C2×C4×D5, C23×D5, D5×C22×C4, Q8.10D10, D48D10, C42.126D10

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

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A ··· 4N 4O ··· 4X 5A 5B 10A ··· 10F 20A ··· 20H 20I ··· 20AF order 1 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 10 ··· 10 2 ··· 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D5 D10 D10 D10 C4×D5 2+ 1+4 2- 1+4 Q8.10D10 D4⋊8D10 kernel C42.126D10 C42⋊D5 C4×D20 D5×C4⋊C4 D20⋊8C4 Q8×Dic5 Q8×C20 C2×Q8⋊2D5 Q8⋊2D5 C4×Q8 C42 C4⋊C4 C2×Q8 Q8 C10 C10 C2 C2 # reps 1 3 3 3 3 1 1 1 16 2 6 6 2 16 1 1 4 4

Matrix representation of C42.126D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 38 35 35 29 0 0 7 3 14 6 0 0 35 29 3 6 0 0 14 6 34 38
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 23 5 0 0 0 0 1 18 0 0 0 0 0 0 23 5 0 0 0 0 1 18
,
 22 13 0 0 0 0 19 0 0 0 0 0 0 0 0 0 6 6 0 0 0 0 34 0 0 0 35 35 0 0 0 0 7 0 0 0
,
 22 9 0 0 0 0 19 19 0 0 0 0 0 0 20 38 0 0 0 0 38 21 0 0 0 0 0 0 20 38 0 0 0 0 38 21

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,38,7,35,14,0,0,35,3,29,6,0,0,35,14,3,34,0,0,29,6,6,38],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,23,1,0,0,0,0,5,18,0,0,0,0,0,0,23,1,0,0,0,0,5,18],[22,19,0,0,0,0,13,0,0,0,0,0,0,0,0,0,35,7,0,0,0,0,35,0,0,0,6,34,0,0,0,0,6,0,0,0],[22,19,0,0,0,0,9,19,0,0,0,0,0,0,20,38,0,0,0,0,38,21,0,0,0,0,0,0,20,38,0,0,0,0,38,21] >;`

C42.126D10 in GAP, Magma, Sage, TeX

`C_4^2._{126}D_{10}`
`% in TeX`

`G:=Group("C4^2.126D10");`
`// GroupNames label`

`G:=SmallGroup(320,1246);`
`// by ID`

`G=gap.SmallGroup(320,1246);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,136,12550]);`
`// Polycyclic`

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

׿
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