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## G = C40.4D4order 320 = 26·5

### 4th non-split extension by C40 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C40.4D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×Dic10 — C2×Dic20 — C40.4D4
 Lower central C5 — C10 — C2×C20 — C40.4D4
 Upper central C1 — C22 — C22×C4 — C2×M4(2)

Generators and relations for C40.4D4
G = < a,b,c | a40=b4=1, c2=a20, bab-1=a19, cac-1=a-1, cbc-1=a20b-1 >

Subgroups: 406 in 110 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C2×C8, M4(2), Q16, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, Q8⋊C4, C4.Q8, C22⋊Q8, C2×M4(2), C2×Q16, C40, C40, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C8.D4, Dic20, C10.D4, C4⋊Dic5, C23.D5, C2×C40, C5×M4(2), C2×Dic10, C22×C20, C20.44D4, C406C4, C2×Dic20, C20.48D4, C10×M4(2), C40.4D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8.C22, D20, C5⋊D4, C22×D5, C8.D4, C2×D20, C4○D20, C2×C5⋊D4, C8.D10, C207D4, C40.4D4

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 D10 D10 C5⋊D4 D20 D20 C4○D20 C8.C22 C8.D10 kernel C40.4D4 C20.44D4 C40⋊6C4 C2×Dic20 C20.48D4 C10×M4(2) C40 C2×C20 C22×C10 C2×M4(2) C20 C2×C8 C22×C4 C8 C2×C4 C23 C4 C10 C2 # reps 1 2 1 1 2 1 2 1 1 2 2 4 2 8 4 4 8 2 8

Matrix representation of C40.4D4 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 37 33 0 0 0 0 20 4 0 0 0 0 0 0 8 16 0 0 0 0 1 33
,
 19 39 0 0 0 0 17 22 0 0 0 0 0 0 0 0 1 0 0 0 0 0 40 40 0 0 1 0 0 0 0 0 40 40 0 0
,
 19 39 0 0 0 0 16 22 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 40 0 0 0 0 0 0 40 0 0

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,20,0,0,0,0,33,4,0,0,0,0,0,0,8,1,0,0,0,0,16,33],[19,17,0,0,0,0,39,22,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,40,0,0,1,40,0,0,0,0,0,40,0,0],[19,16,0,0,0,0,39,22,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C40.4D4 in GAP, Magma, Sage, TeX

`C_{40}._4D_4`
`% in TeX`

`G:=Group("C40.4D4");`
`// GroupNames label`

`G:=SmallGroup(320,764);`
`// by ID`

`G=gap.SmallGroup(320,764);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,344,254,387,1684,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^4=1,c^2=a^20,b*a*b^-1=a^19,c*a*c^-1=a^-1,c*b*c^-1=a^20*b^-1>;`
`// generators/relations`

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