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## G = C40⋊5D6order 480 = 25·3·5

### 5th semidirect product of C40 and D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C40⋊5D6
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — C20⋊D6 — C40⋊5D6
 Lower central C15 — C30 — C60 — C40⋊5D6
 Upper central C1 — C2 — C4 — C8

Generators and relations for C405D6
G = < a,b,c | a40=b6=c2=1, bab-1=a-1, cac=a9, cbc=b-1 >

Subgroups: 1180 in 152 conjugacy classes, 40 normal (26 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22 [×9], C5, S3 [×4], C6, C6 [×2], C8, C8, C2×C4, D4 [×6], C23 [×2], D5 [×4], C10, C10 [×2], Dic3, C12, D6 [×7], C2×C6 [×2], C15, C2×C8, D8 [×4], C2×D4 [×2], Dic5, C20, D10 [×7], C2×C10 [×2], C3⋊C8, C24, C4×S3, D12 [×2], C3⋊D4 [×2], C3×D4 [×2], C22×S3 [×2], C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, C2×D8, C52C8, C40, C4×D5, D20 [×2], C5⋊D4 [×2], C5×D4 [×2], C22×D5 [×2], S3×C8, D24, D4⋊S3 [×2], C3×D8, S3×D4 [×2], Dic15, C60, S3×D5 [×4], C6×D5 [×2], S3×C10 [×2], D30, C8×D5, D40, D4⋊D5 [×2], C5×D8, D4×D5 [×2], S3×D8, C153C8, C120, C15⋊D4 [×2], C3×D20 [×2], C5×D12 [×2], C4×D15, C2×S3×D5 [×2], D5×D8, C15⋊D8 [×2], C3×D40, C5×D24, C8×D15, C20⋊D6 [×2], C405D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], D8 [×2], C2×D4, D10 [×3], C22×S3, C2×D8, C22×D5, S3×D4, S3×D5, D4×D5, S3×D8, C2×S3×D5, D5×D8, C20⋊D6, C405D6

Smallest permutation representation of C405D6
On 120 points
Generators in S120
```(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)
(1 46 109 16 71 84)(2 45 110 15 72 83)(3 44 111 14 73 82)(4 43 112 13 74 81)(5 42 113 12 75 120)(6 41 114 11 76 119)(7 80 115 10 77 118)(8 79 116 9 78 117)(17 70 85 40 47 108)(18 69 86 39 48 107)(19 68 87 38 49 106)(20 67 88 37 50 105)(21 66 89 36 51 104)(22 65 90 35 52 103)(23 64 91 34 53 102)(24 63 92 33 54 101)(25 62 93 32 55 100)(26 61 94 31 56 99)(27 60 95 30 57 98)(28 59 96 29 58 97)
(1 89)(2 98)(3 107)(4 116)(5 85)(6 94)(7 103)(8 112)(9 81)(10 90)(11 99)(12 108)(13 117)(14 86)(15 95)(16 104)(17 113)(18 82)(19 91)(20 100)(21 109)(22 118)(23 87)(24 96)(25 105)(26 114)(27 83)(28 92)(29 101)(30 110)(31 119)(32 88)(33 97)(34 106)(35 115)(36 84)(37 93)(38 102)(39 111)(40 120)(41 61)(42 70)(43 79)(44 48)(45 57)(46 66)(47 75)(49 53)(50 62)(51 71)(52 80)(54 58)(55 67)(56 76)(59 63)(60 72)(64 68)(65 77)(69 73)(74 78)```

`G:=sub<Sym(120)| (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), (1,46,109,16,71,84)(2,45,110,15,72,83)(3,44,111,14,73,82)(4,43,112,13,74,81)(5,42,113,12,75,120)(6,41,114,11,76,119)(7,80,115,10,77,118)(8,79,116,9,78,117)(17,70,85,40,47,108)(18,69,86,39,48,107)(19,68,87,38,49,106)(20,67,88,37,50,105)(21,66,89,36,51,104)(22,65,90,35,52,103)(23,64,91,34,53,102)(24,63,92,33,54,101)(25,62,93,32,55,100)(26,61,94,31,56,99)(27,60,95,30,57,98)(28,59,96,29,58,97), (1,89)(2,98)(3,107)(4,116)(5,85)(6,94)(7,103)(8,112)(9,81)(10,90)(11,99)(12,108)(13,117)(14,86)(15,95)(16,104)(17,113)(18,82)(19,91)(20,100)(21,109)(22,118)(23,87)(24,96)(25,105)(26,114)(27,83)(28,92)(29,101)(30,110)(31,119)(32,88)(33,97)(34,106)(35,115)(36,84)(37,93)(38,102)(39,111)(40,120)(41,61)(42,70)(43,79)(44,48)(45,57)(46,66)(47,75)(49,53)(50,62)(51,71)(52,80)(54,58)(55,67)(56,76)(59,63)(60,72)(64,68)(65,77)(69,73)(74,78)>;`

`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), (1,46,109,16,71,84)(2,45,110,15,72,83)(3,44,111,14,73,82)(4,43,112,13,74,81)(5,42,113,12,75,120)(6,41,114,11,76,119)(7,80,115,10,77,118)(8,79,116,9,78,117)(17,70,85,40,47,108)(18,69,86,39,48,107)(19,68,87,38,49,106)(20,67,88,37,50,105)(21,66,89,36,51,104)(22,65,90,35,52,103)(23,64,91,34,53,102)(24,63,92,33,54,101)(25,62,93,32,55,100)(26,61,94,31,56,99)(27,60,95,30,57,98)(28,59,96,29,58,97), (1,89)(2,98)(3,107)(4,116)(5,85)(6,94)(7,103)(8,112)(9,81)(10,90)(11,99)(12,108)(13,117)(14,86)(15,95)(16,104)(17,113)(18,82)(19,91)(20,100)(21,109)(22,118)(23,87)(24,96)(25,105)(26,114)(27,83)(28,92)(29,101)(30,110)(31,119)(32,88)(33,97)(34,106)(35,115)(36,84)(37,93)(38,102)(39,111)(40,120)(41,61)(42,70)(43,79)(44,48)(45,57)(46,66)(47,75)(49,53)(50,62)(51,71)(52,80)(54,58)(55,67)(56,76)(59,63)(60,72)(64,68)(65,77)(69,73)(74,78) );`

`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)], [(1,46,109,16,71,84),(2,45,110,15,72,83),(3,44,111,14,73,82),(4,43,112,13,74,81),(5,42,113,12,75,120),(6,41,114,11,76,119),(7,80,115,10,77,118),(8,79,116,9,78,117),(17,70,85,40,47,108),(18,69,86,39,48,107),(19,68,87,38,49,106),(20,67,88,37,50,105),(21,66,89,36,51,104),(22,65,90,35,52,103),(23,64,91,34,53,102),(24,63,92,33,54,101),(25,62,93,32,55,100),(26,61,94,31,56,99),(27,60,95,30,57,98),(28,59,96,29,58,97)], [(1,89),(2,98),(3,107),(4,116),(5,85),(6,94),(7,103),(8,112),(9,81),(10,90),(11,99),(12,108),(13,117),(14,86),(15,95),(16,104),(17,113),(18,82),(19,91),(20,100),(21,109),(22,118),(23,87),(24,96),(25,105),(26,114),(27,83),(28,92),(29,101),(30,110),(31,119),(32,88),(33,97),(34,106),(35,115),(36,84),(37,93),(38,102),(39,111),(40,120),(41,61),(42,70),(43,79),(44,48),(45,57),(46,66),(47,75),(49,53),(50,62),(51,71),(52,80),(54,58),(55,67),(56,76),(59,63),(60,72),(64,68),(65,77),(69,73),(74,78)])`

51 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 12 15A 15B 20A 20B 24A 24B 30A 30B 40A 40B 40C 40D 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 2 2 2 2 3 4 4 5 5 6 6 6 8 8 8 8 10 10 10 10 10 10 12 15 15 20 20 24 24 30 30 40 40 40 40 60 60 60 60 120 ··· 120 size 1 1 12 12 15 15 20 20 2 2 30 2 2 2 40 40 2 2 30 30 2 2 24 24 24 24 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D8 D10 D10 S3×D4 S3×D5 D4×D5 S3×D8 C2×S3×D5 D5×D8 C20⋊D6 C40⋊5D6 kernel C40⋊5D6 C15⋊D8 C3×D40 C5×D24 C8×D15 C20⋊D6 D40 Dic15 D30 D24 C40 D20 D15 C24 D12 C10 C8 C6 C5 C4 C3 C2 C1 # reps 1 2 1 1 1 2 1 1 1 2 1 2 4 2 4 1 2 2 2 2 4 4 8

Matrix representation of C405D6 in GL6(𝔽241)

 0 190 0 0 0 0 52 52 0 0 0 0 0 0 0 22 0 0 0 0 230 22 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 52 51 0 0 0 0 188 189 0 0 0 0 0 0 0 22 0 0 0 0 11 0 0 0 0 0 0 0 0 1 0 0 0 0 240 1
,
 189 190 0 0 0 0 53 52 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 240 0 0 0 0 0 240

`G:=sub<GL(6,GF(241))| [0,52,0,0,0,0,190,52,0,0,0,0,0,0,0,230,0,0,0,0,22,22,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[52,188,0,0,0,0,51,189,0,0,0,0,0,0,0,11,0,0,0,0,22,0,0,0,0,0,0,0,0,240,0,0,0,0,1,1],[189,53,0,0,0,0,190,52,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,240,240] >;`

C405D6 in GAP, Magma, Sage, TeX

`C_{40}\rtimes_5D_6`
`% in TeX`

`G:=Group("C40:5D6");`
`// GroupNames label`

`G:=SmallGroup(480,332);`
`// by ID`

`G=gap.SmallGroup(480,332);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,135,142,675,346,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^6=c^2=1,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^-1>;`
`// generators/relations`

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