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## G = C42.115D10order 320 = 26·5

### 115th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.115D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C42⋊D5 — C42.115D10
 Lower central C5 — C2×C10 — C42.115D10
 Upper central C1 — C22 — C4×D4

Generators and relations for C42.115D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=b2c-1 >

Subgroups: 718 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×11], D4 [×4], Q8 [×4], C23 [×2], C23, D5, C10 [×3], C10 [×2], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4, C2×D4, C2×D4 [×2], C2×Q8 [×3], Dic5 [×7], C20 [×2], C20 [×4], D10 [×3], C2×C10, C2×C10 [×6], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×3], C422C2 [×2], C4⋊Q8, Dic10 [×4], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C5⋊D4 [×2], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×2], C22×D5, C22×C10 [×2], C22.36C24, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×4], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×2], C23.D5 [×6], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×Dic10 [×2], C2×C4×D5, C2×C5⋊D4 [×2], C22×C20 [×2], D4×C10, C4×Dic10, C42⋊D5, C23.D10 [×2], Dic5.5D4 [×2], C20⋊Q8, D102Q8, C20.48D4 [×2], C23.23D10 [×2], C20.17D4, C202D4, D4×C20, C42.115D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5 [×7], C22.36C24, C4○D20 [×2], C23×D5, C2×C4○D20, D46D10, D4.10D10, C42.115D10

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 D10 D10 C4○D20 2+ 1+4 2- 1+4 D4⋊6D10 D4.10D10 kernel C42.115D10 C4×Dic10 C42⋊D5 C23.D10 Dic5.5D4 C20⋊Q8 D10⋊2Q8 C20.48D4 C23.23D10 C20.17D4 C20⋊2D4 D4×C20 C4×D4 C20 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C4 C10 C10 C2 C2 # reps 1 1 1 2 2 1 1 2 2 1 1 1 2 4 2 4 2 4 2 16 1 1 4 4

Matrix representation of C42.115D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 37 31 22 21 0 0 10 15 20 19 0 0 4 10 4 10 0 0 31 26 31 26
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 2 28 0 0 0 0 13 39 0 0 0 0 0 0 2 28 0 0 0 0 13 39
,
 40 39 0 0 0 0 0 1 0 0 0 0 0 0 1 6 2 12 0 0 35 6 29 12 0 0 0 0 40 35 0 0 0 0 6 35
,
 1 2 0 0 0 0 40 40 0 0 0 0 0 0 40 0 39 0 0 0 6 1 12 2 0 0 1 0 1 0 0 0 35 40 35 40

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,37,10,4,31,0,0,31,15,10,26,0,0,22,20,4,31,0,0,21,19,10,26],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,2,13,0,0,0,0,28,39,0,0,0,0,0,0,2,13,0,0,0,0,28,39],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,1,35,0,0,0,0,6,6,0,0,0,0,2,29,40,6,0,0,12,12,35,35],[1,40,0,0,0,0,2,40,0,0,0,0,0,0,40,6,1,35,0,0,0,1,0,40,0,0,39,12,1,35,0,0,0,2,0,40] >;`

C42.115D10 in GAP, Magma, Sage, TeX

`C_4^2._{115}D_{10}`
`% in TeX`

`G:=Group("C4^2.115D10");`
`// GroupNames label`

`G:=SmallGroup(320,1233);`
`// by ID`

`G=gap.SmallGroup(320,1233);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,100,675,570,12550]);`
`// Polycyclic`

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

׿
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