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## G = C42.96D10order 320 = 26·5

### 96th non-split extension by C42 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42.96D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C5⋊D4 — Dic5⋊4D4 — C42.96D10
 Lower central C5 — C2×C10 — C42.96D10
 Upper central C1 — C22 — C42⋊C2

Generators and relations for C42.96D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=b2c-1 >

Subgroups: 662 in 214 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×14], C22, C22 [×2], C22 [×5], C5, C2×C4 [×6], C2×C4 [×15], D4 [×2], Q8 [×2], C23, C23, D5, C10 [×3], C10 [×2], C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×14], C22×C4, C22×C4 [×3], C2×D4, C2×Q8, Dic5 [×2], Dic5 [×6], C20 [×6], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4, C42⋊C2, C42⋊C2 [×2], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×3], C422C2 [×2], Dic10 [×2], C4×D5 [×2], C2×Dic5 [×7], C2×Dic5 [×4], C5⋊D4 [×2], C2×C20 [×6], C2×C20 [×2], C22×D5, C22×C10, C22.46C24, C4×Dic5 [×3], C10.D4 [×11], C4⋊Dic5 [×3], D10⋊C4 [×5], C23.D5, C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5, C22×Dic5 [×2], C2×C5⋊D4, C22×C20, C4×Dic10, C20.6Q8, C42⋊D5, C422D5, C23.11D10, Dic5.14D4, Dic54D4, C22.D20, Dic5.Q8 [×2], D10⋊Q8, C4⋊C4⋊D5, C2×C10.D4, C23.23D10, C5×C42⋊C2, C42.96D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2- 1+4, C22×D5 [×7], C22.46C24, C4○D20 [×2], C23×D5, C2×C4○D20, D5×C4○D4, D4.10D10, C42.96D10

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

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N ··· 4R 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20AB order 1 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 20 2 ··· 2 4 4 4 10 10 10 10 20 ··· 20 2 2 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 D10 C4○D20 2- 1+4 D5×C4○D4 D4.10D10 kernel C42.96D10 C4×Dic10 C20.6Q8 C42⋊D5 C42⋊2D5 C23.11D10 Dic5.14D4 Dic5⋊4D4 C22.D20 Dic5.Q8 D10⋊Q8 C4⋊C4⋊D5 C2×C10.D4 C23.23D10 C5×C42⋊C2 C42⋊C2 Dic5 C2×C10 C42 C22⋊C4 C4⋊C4 C22×C4 C22 C10 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2 4 4 4 4 4 2 16 1 4 4

Matrix representation of C42.96D10 in GL4(𝔽41) generated by

 9 0 0 0 0 9 0 0 0 0 40 0 0 0 40 1
,
 18 35 0 0 6 23 0 0 0 0 32 0 0 0 0 32
,
 6 6 0 0 35 1 0 0 0 0 1 39 0 0 0 40
,
 6 6 0 0 1 35 0 0 0 0 40 2 0 0 40 1
`G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,40,40,0,0,0,1],[18,6,0,0,35,23,0,0,0,0,32,0,0,0,0,32],[6,35,0,0,6,1,0,0,0,0,1,0,0,0,39,40],[6,1,0,0,6,35,0,0,0,0,40,40,0,0,2,1] >;`

C42.96D10 in GAP, Magma, Sage, TeX

`C_4^2._{96}D_{10}`
`% in TeX`

`G:=Group("C4^2.96D10");`
`// GroupNames label`

`G:=SmallGroup(320,1203);`
`// by ID`

`G=gap.SmallGroup(320,1203);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,100,675,136,12550]);`
`// 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*b^2,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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