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## G = C42.214D10order 320 = 26·5

### 34th non-split extension by C42 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C42.214D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C4.D20 — C42.214D10
 Lower central C5 — C10 — C2×C20 — C42.214D10
 Upper central C1 — C22 — C42 — C4.4D4

Generators and relations for C42.214D10
G = < a,b,c,d | a4=b4=c10=1, d2=cbc-1=b-1, ab=ba, cac-1=a-1b2, ad=da, bd=db, dcd-1=b-1c-1 >

Subgroups: 542 in 130 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×4], Q8 [×4], C23 [×2], D5, C10, C10 [×2], C10, C42, C22⋊C4 [×4], C2×C8 [×2], D8 [×2], SD16 [×4], Q16 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8, Dic5, C20 [×2], C20 [×3], D10 [×3], C2×C10, C2×C10 [×3], C4×C8, C4.4D4, C4.4D4, C2×D8, C2×SD16 [×2], C2×Q16, C52C8 [×4], Dic10 [×2], D20 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×C10, C8.12D4, C2×C52C8 [×2], D10⋊C4 [×2], D4⋊D5 [×2], D4.D5 [×2], Q8⋊D5 [×2], C5⋊Q16 [×2], C4×C20, C5×C22⋊C4 [×2], C2×Dic10, C2×D20, D4×C10, Q8×C10, C4×C52C8, C4.D20, C2×D4⋊D5, C2×D4.D5, C2×Q8⋊D5, C2×C5⋊Q16, C5×C4.4D4, C42.214D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C41D4, C4○D8 [×2], C5⋊D4 [×2], C22×D5, C8.12D4, D4×D5 [×2], C2×C5⋊D4, C20⋊D4, D4.8D10 [×2], C42.214D10

Smallest permutation representation of C42.214D10
On 160 points
Generators in S160
```(1 143 40 125)(2 101 31 138)(3 145 32 127)(4 103 33 140)(5 147 34 129)(6 105 35 132)(7 149 36 121)(8 107 37 134)(9 141 38 123)(10 109 39 136)(11 106 69 133)(12 150 70 122)(13 108 61 135)(14 142 62 124)(15 110 63 137)(16 144 64 126)(17 102 65 139)(18 146 66 128)(19 104 67 131)(20 148 68 130)(21 152 72 90)(22 46 73 119)(23 154 74 82)(24 48 75 111)(25 156 76 84)(26 50 77 113)(27 158 78 86)(28 42 79 115)(29 160 80 88)(30 44 71 117)(41 95 114 51)(43 97 116 53)(45 99 118 55)(47 91 120 57)(49 93 112 59)(52 159 96 87)(54 151 98 89)(56 153 100 81)(58 155 92 83)(60 157 94 85)
(1 73 63 56)(2 57 64 74)(3 75 65 58)(4 59 66 76)(5 77 67 60)(6 51 68 78)(7 79 69 52)(8 53 70 80)(9 71 61 54)(10 55 62 72)(11 96 36 28)(12 29 37 97)(13 98 38 30)(14 21 39 99)(15 100 40 22)(16 23 31 91)(17 92 32 24)(18 25 33 93)(19 94 34 26)(20 27 35 95)(41 130 86 105)(42 106 87 121)(43 122 88 107)(44 108 89 123)(45 124 90 109)(46 110 81 125)(47 126 82 101)(48 102 83 127)(49 128 84 103)(50 104 85 129)(111 139 155 145)(112 146 156 140)(113 131 157 147)(114 148 158 132)(115 133 159 149)(116 150 160 134)(117 135 151 141)(118 142 152 136)(119 137 153 143)(120 144 154 138)
(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 56 72 63 62 73 55)(2 54 74 61 64 71 57 9)(3 8 58 80 65 70 75 53)(4 52 76 69 66 79 59 7)(5 6 60 78 67 68 77 51)(11 18 28 93 36 33 96 25)(12 24 97 32 37 92 29 17)(13 16 30 91 38 31 98 23)(14 22 99 40 39 100 21 15)(19 20 26 95 34 35 94 27)(41 147 105 157 86 131 130 113)(42 112 121 140 87 156 106 146)(43 145 107 155 88 139 122 111)(44 120 123 138 89 154 108 144)(45 143 109 153 90 137 124 119)(46 118 125 136 81 152 110 142)(47 141 101 151 82 135 126 117)(48 116 127 134 83 160 102 150)(49 149 103 159 84 133 128 115)(50 114 129 132 85 158 104 148)```

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 C4○D8 C5⋊D4 D4×D5 D4.8D10 kernel C42.214D10 C4×C5⋊2C8 C4.D20 C2×D4⋊D5 C2×D4.D5 C2×Q8⋊D5 C2×C5⋊Q16 C5×C4.4D4 C5⋊2C8 C2×C20 C4.4D4 C42 C2×D4 C2×Q8 C10 C2×C4 C4 C2 # reps 1 1 1 1 1 1 1 1 4 2 2 2 2 2 8 8 4 8

Matrix representation of C42.214D10 in GL6(𝔽41)

 32 9 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 9 18 0 0 0 0 32 32
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 39 0 0 0 0 1 1
,
 40 0 0 0 0 0 39 1 0 0 0 0 0 0 24 38 0 0 0 0 3 3 0 0 0 0 0 0 0 17 0 0 0 0 29 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 17 1 0 0 0 0 38 24 0 0 0 0 0 0 24 24 0 0 0 0 29 0

`G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,9,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,32,0,0,0,0,18,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[40,39,0,0,0,0,0,1,0,0,0,0,0,0,24,3,0,0,0,0,38,3,0,0,0,0,0,0,0,29,0,0,0,0,17,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,17,38,0,0,0,0,1,24,0,0,0,0,0,0,24,29,0,0,0,0,24,0] >;`

C42.214D10 in GAP, Magma, Sage, TeX

`C_4^2._{214}D_{10}`
`% in TeX`

`G:=Group("C4^2.214D10");`
`// GroupNames label`

`G:=SmallGroup(320,686);`
`// by ID`

`G=gap.SmallGroup(320,686);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,219,1123,297,136,12550]);`
`// Polycyclic`

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

׿
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