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## G = C40.8D4order 320 = 26·5

### 8th non-split extension by C40 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C40.8D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C40 — C2×Dic20 — C40.8D4
 Lower central C5 — C10 — C20 — C40 — C40.8D4
 Upper central C1 — C2 — C2×C4 — C2×C8 — C8.C4

Generators and relations for C40.8D4
G = < a,b,c | a40=1, b4=a20, c2=a5, bab-1=a31, cac-1=a9, cbc-1=a5b3 >

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + - + + - image C1 C2 C2 C2 C4 D4 D4 D5 D8 SD16 D10 C4×D5 D20 C5⋊D4 C8.17D4 D4⋊D5 Q8⋊D5 C40.8D4 kernel C40.8D4 C20.4C8 C5×C8.C4 C2×Dic20 Dic20 C40 C2×C20 C8.C4 C20 C2×C10 C2×C8 C8 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 1 1 4 1 1 2 2 2 2 4 4 4 2 2 2 8

Matrix representation of C40.8D4 in GL4(𝔽241) generated by

 185 14 0 0 218 208 0 0 0 0 214 47 0 0 181 33
,
 0 0 1 0 0 0 0 1 44 156 0 0 88 197 0 0
,
 0 0 207 86 0 0 51 34 73 176 0 0 82 168 0 0
`G:=sub<GL(4,GF(241))| [185,218,0,0,14,208,0,0,0,0,214,181,0,0,47,33],[0,0,44,88,0,0,156,197,1,0,0,0,0,1,0,0],[0,0,73,82,0,0,176,168,207,51,0,0,86,34,0,0] >;`

C40.8D4 in GAP, Magma, Sage, TeX

`C_{40}._8D_4`
`% in TeX`

`G:=Group("C40.8D4");`
`// GroupNames label`

`G:=SmallGroup(320,54);`
`// by ID`

`G=gap.SmallGroup(320,54);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,758,184,675,794,192,1684,851,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=1,b^4=a^20,c^2=a^5,b*a*b^-1=a^31,c*a*c^-1=a^9,c*b*c^-1=a^5*b^3>;`
`// generators/relations`

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