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## G = C40.29D4order 320 = 26·5

### 29th non-split extension by C40 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C40.29D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D20.3C4 — C40.29D4
 Lower central C5 — C10 — C2×C20 — C40.29D4
 Upper central C1 — C2 — C2×C4 — C2×Q16

Generators and relations for C40.29D4
G = < a,b,c | a40=c2=1, b4=a20, bab-1=a-1, cac=a9, cbc=b3 >

Subgroups: 334 in 100 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C2×C8, C2×C8, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, C52C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×Q8, D4.5D4, C8×D5, C8⋊D5, C4.Dic5, C4.Dic5, Q8⋊D5, C5⋊Q16, C2×C40, C5×Q16, C4○D20, Q8×C10, C40.6C4, C20.10D4, D20.3C4, C20.C23, C10×Q16, C40.29D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C5⋊D4, C22×D5, D4.5D4, D4×D5, D42D5, C2×C5⋊D4, C202D4, C40.29D4

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 D10 D10 C5⋊D4 D4.5D4 D4×D5 D4⋊2D5 C40.29D4 kernel C40.29D4 C40.6C4 C20.10D4 D20.3C4 C20.C23 C10×Q16 C40 Dic10 D20 C2×Q16 C2×C10 C2×C8 C2×Q8 C8 C5 C4 C22 C1 # reps 1 1 2 1 2 1 2 1 1 2 2 2 4 8 2 2 2 8

Matrix representation of C40.29D4 in GL4(𝔽41) generated by

 22 20 31 21 0 0 31 40 31 2 4 27 19 21 21 19
,
 31 7 6 32 9 20 11 12 32 24 15 39 31 40 21 16
,
 6 40 6 40 35 35 0 35 0 0 34 8 0 0 35 7
`G:=sub<GL(4,GF(41))| [22,0,31,19,20,0,2,21,31,31,4,21,21,40,27,19],[31,9,32,31,7,20,24,40,6,11,15,21,32,12,39,16],[6,35,0,0,40,35,0,0,6,0,34,35,40,35,8,7] >;`

C40.29D4 in GAP, Magma, Sage, TeX

`C_{40}._{29}D_4`
`% in TeX`

`G:=Group("C40.29D4");`
`// GroupNames label`

`G:=SmallGroup(320,819);`
`// by ID`

`G=gap.SmallGroup(320,819);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,184,1123,297,136,438,102,12550]);`
`// Polycyclic`

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

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