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## G = C40.91D4order 320 = 26·5

### 14th non-split extension by C40 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C40.91D4
 Chief series C1 — C5 — C10 — C20 — C40 — C2×C40 — C2×C5⋊2C16 — C40.91D4
 Lower central C5 — C10 — C40.91D4
 Upper central C1 — C2×C8 — C22×C8

Generators and relations for C40.91D4
G = < a,b,c | a40=1, b4=a10, c2=a25, bab-1=cac-1=a9, cbc-1=a15b3 >

Subgroups: 118 in 66 conjugacy classes, 39 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, C10, C10, C16, C2×C8, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C2×C16, C22×C8, C40, C40, C2×C20, C2×C20, C22×C10, C22⋊C16, C52C16, C2×C40, C2×C40, C22×C20, C2×C52C16, C22×C40, C40.91D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D5, C16, C22⋊C4, C2×C8, M4(2), Dic5, D10, C22⋊C8, C2×C16, M5(2), C52C8, C2×Dic5, C5⋊D4, C22⋊C16, C52C16, C2×C52C8, C4.Dic5, C23.D5, C2×C52C16, C20.4C8, C20.55D4, C40.91D4

Smallest permutation representation of C40.91D4
On 160 points
Generators in S160
```(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 57 139 103 11 67 149 113 21 77 159 83 31 47 129 93)(2 66 140 112 12 76 150 82 22 46 160 92 32 56 130 102)(3 75 141 81 13 45 151 91 23 55 121 101 33 65 131 111)(4 44 142 90 14 54 152 100 24 64 122 110 34 74 132 120)(5 53 143 99 15 63 153 109 25 73 123 119 35 43 133 89)(6 62 144 108 16 72 154 118 26 42 124 88 36 52 134 98)(7 71 145 117 17 41 155 87 27 51 125 97 37 61 135 107)(8 80 146 86 18 50 156 96 28 60 126 106 38 70 136 116)(9 49 147 95 19 59 157 105 29 69 127 115 39 79 137 85)(10 58 148 104 20 68 158 114 30 78 128 84 40 48 138 94)
(1 118 26 103 11 88 36 113 21 98 6 83 31 108 16 93)(2 87 27 112 12 97 37 82 22 107 7 92 32 117 17 102)(3 96 28 81 13 106 38 91 23 116 8 101 33 86 18 111)(4 105 29 90 14 115 39 100 24 85 9 110 34 95 19 120)(5 114 30 99 15 84 40 109 25 94 10 119 35 104 20 89)(41 155 66 140 51 125 76 150 61 135 46 160 71 145 56 130)(42 124 67 149 52 134 77 159 62 144 47 129 72 154 57 139)(43 133 68 158 53 143 78 128 63 153 48 138 73 123 58 148)(44 142 69 127 54 152 79 137 64 122 49 147 74 132 59 157)(45 151 70 136 55 121 80 146 65 131 50 156 75 141 60 126)```

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

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

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

104 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5A 5B 8A ··· 8H 8I 8J 8K 8L 10A ··· 10N 16A ··· 16P 20A ··· 20P 40A ··· 40AF order 1 2 2 2 2 2 4 4 4 4 4 4 5 5 8 ··· 8 8 8 8 8 10 ··· 10 16 ··· 16 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 1 1 1 1 2 2 2 2 1 ··· 1 2 2 2 2 2 ··· 2 10 ··· 10 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C4 C4 C8 C8 C16 D4 D5 M4(2) Dic5 D10 Dic5 M5(2) C5⋊D4 C5⋊2C8 C5⋊2C8 C4.Dic5 C5⋊2C16 C20.4C8 kernel C40.91D4 C2×C5⋊2C16 C22×C40 C2×C40 C22×C20 C2×C20 C22×C10 C2×C10 C40 C22×C8 C20 C2×C8 C2×C8 C22×C4 C10 C8 C2×C4 C23 C4 C22 C2 # reps 1 2 1 2 2 4 4 16 2 2 2 2 2 2 4 8 4 4 8 16 16

Matrix representation of C40.91D4 in GL3(𝔽241) generated by

 233 0 0 0 116 0 0 0 41
,
 76 0 0 0 0 1 0 30 0
,
 165 0 0 0 0 1 0 211 0
`G:=sub<GL(3,GF(241))| [233,0,0,0,116,0,0,0,41],[76,0,0,0,0,30,0,1,0],[165,0,0,0,0,211,0,1,0] >;`

C40.91D4 in GAP, Magma, Sage, TeX

`C_{40}._{91}D_4`
`% in TeX`

`G:=Group("C40.91D4");`
`// GroupNames label`

`G:=SmallGroup(320,107);`
`// by ID`

`G=gap.SmallGroup(320,107);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,100,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=1,b^4=a^10,c^2=a^25,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=a^15*b^3>;`
`// generators/relations`

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