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## G = C40.22D4order 320 = 26·5

### 22nd non-split extension by C40 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C40.22D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — C20.17D4 — C40.22D4
 Lower central C5 — C10 — C2×C20 — C40.22D4
 Upper central C1 — C22 — C2×C4 — C2×D8

Generators and relations for C40.22D4
G = < a,b,c | a40=b4=1, c2=a20, bab-1=a9, cac-1=a-1, cbc-1=a20b-1 >

Subgroups: 494 in 130 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×4], D4 [×4], Q8 [×4], C23 [×2], C10, C10 [×2], C10 [×2], C42, C22⋊C4 [×4], C2×C8, C2×C8, D8 [×2], SD16 [×4], Q16 [×2], C2×D4 [×2], C2×Q8 [×2], Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×6], C4×C8, C4.4D4 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C52C8 [×2], C40 [×2], Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C5×D4 [×4], C22×C10 [×2], C8.12D4, Dic20 [×2], C2×C52C8, C4×Dic5, D4.D5 [×4], C23.D5 [×4], C2×C40, C5×D8 [×2], C2×Dic10 [×2], D4×C10 [×2], C8×Dic5, C2×Dic20, C2×D4.D5 [×2], C20.17D4 [×2], C10×D8, C40.22D4
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, D83D5 [×2], C20⋊D4, C40.22D4

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

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 10G ··· 10N 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 10 ··· 10 10 ··· 10 20 20 20 20 40 ··· 40 size 1 1 1 1 8 8 2 2 10 10 10 10 40 40 2 2 2 2 2 2 10 10 10 10 2 ··· 2 8 ··· 8 4 4 4 4 4 ··· 4

50 irreducible representations

 dim 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 D4 D4 D4 D5 D10 D10 C4○D8 C5⋊D4 D4×D5 D4×D5 D8⋊3D5 kernel C40.22D4 C8×Dic5 C2×Dic20 C2×D4.D5 C20.17D4 C10×D8 C5⋊2C8 C40 C2×Dic5 C2×D8 C2×C8 C2×D4 C10 C8 C4 C22 C2 # reps 1 1 1 2 2 1 2 2 2 2 2 4 8 8 2 2 8

Matrix representation of C40.22D4 in GL4(𝔽41) generated by

 37 8 0 0 0 10 0 0 0 0 29 12 0 0 29 29
,
 9 0 0 0 26 32 0 0 0 0 32 0 0 0 0 32
,
 32 22 0 0 15 9 0 0 0 0 32 0 0 0 0 9
`G:=sub<GL(4,GF(41))| [37,0,0,0,8,10,0,0,0,0,29,29,0,0,12,29],[9,26,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[32,15,0,0,22,9,0,0,0,0,32,0,0,0,0,9] >;`

C40.22D4 in GAP, Magma, Sage, TeX

`C_{40}._{22}D_4`
`% in TeX`

`G:=Group("C40.22D4");`
`// GroupNames label`

`G:=SmallGroup(320,782);`
`// by ID`

`G=gap.SmallGroup(320,782);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,1094,135,570,297,136,12550]);`
`// Polycyclic`

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

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