Copied to
clipboard

## G = C40.43D4order 320 = 26·5

### 43rd non-split extension by C40 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C40.43D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — C20.17D4 — C40.43D4
 Lower central C5 — C10 — C2×C20 — C40.43D4
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for C40.43D4
G = < a,b,c | a40=b4=1, c2=a20, bab-1=a9, cac-1=a19, cbc-1=a20b-1 >

Subgroups: 558 in 130 conjugacy classes, 43 normal (31 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], D5, C10, C10 [×2], C10, C42, C22⋊C4 [×4], C2×C8, C2×C8, D8 [×2], SD16 [×4], Q16 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8, Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], C4×C8, C4.4D4 [×2], C2×D8, C2×SD16, C2×SD16, C2×Q16, C52C8 [×2], C40 [×2], Dic10 [×2], D20 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×C10, C8.12D4, C40⋊C2 [×2], C2×C52C8, C4×Dic5, D10⋊C4 [×2], D4⋊D5 [×2], C5⋊Q16 [×2], C23.D5 [×2], C2×C40, C5×SD16 [×2], C2×Dic10, C2×D20, D4×C10, Q8×C10, C8×Dic5, C2×C40⋊C2, C2×D4⋊D5, C20.17D4, C2×C5⋊Q16, C20.23D4, C10×SD16, C40.43D4
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, SD163D5 [×2], C20⋊D4, C40.43D4

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

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

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

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

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 10H 10I 10J 20A 20B 20C 20D 20E 20F 20G 20H 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 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 1 1 8 40 2 2 8 10 10 10 10 40 2 2 2 2 2 2 10 10 10 10 2 ··· 2 8 8 8 8 4 4 4 4 8 8 8 8 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 C4○D8 C5⋊D4 D4×D5 D4×D5 SD16⋊3D5 kernel C40.43D4 C8×Dic5 C2×C40⋊C2 C2×D4⋊D5 C20.17D4 C2×C5⋊Q16 C20.23D4 C10×SD16 C5⋊2C8 C40 C2×Dic5 C2×SD16 C2×C8 C2×D4 C2×Q8 C10 C8 C4 C22 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 8 8 2 2 8

Matrix representation of C40.43D4 in GL6(𝔽41)

 25 36 0 0 0 0 35 16 0 0 0 0 0 0 35 7 0 0 0 0 35 0 0 0 0 0 0 0 30 30 0 0 0 0 26 0
,
 25 36 0 0 0 0 35 16 0 0 0 0 0 0 35 40 0 0 0 0 35 6 0 0 0 0 0 0 32 0 0 0 0 0 0 32
,
 25 36 0 0 0 0 10 16 0 0 0 0 0 0 35 40 0 0 0 0 35 6 0 0 0 0 0 0 11 11 0 0 0 0 15 30

`G:=sub<GL(6,GF(41))| [25,35,0,0,0,0,36,16,0,0,0,0,0,0,35,35,0,0,0,0,7,0,0,0,0,0,0,0,30,26,0,0,0,0,30,0],[25,35,0,0,0,0,36,16,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[25,10,0,0,0,0,36,16,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,11,15,0,0,0,0,11,30] >;`

C40.43D4 in GAP, Magma, Sage, TeX

`C_{40}._{43}D_4`
`% in TeX`

`G:=Group("C40.43D4");`
`// GroupNames label`

`G:=SmallGroup(320,795);`
`// by ID`

`G=gap.SmallGroup(320,795);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,1094,135,184,570,297,136,12550]);`
`// 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^19,c*b*c^-1=a^20*b^-1>;`
`// generators/relations`

׿
×
𝔽