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## G = C60.29D4order 480 = 25·3·5

### 29th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C60.29D4
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C3×C4.Dic5 — C60.29D4
 Lower central C15 — C30 — C2×C30 — C60.29D4
 Upper central C1 — C2 — C2×C4

Generators and relations for C60.29D4
G = < a,b,c | a60=c2=1, b4=a30, bab-1=a19, cac=a-1, cbc=a45b3 >

Subgroups: 764 in 92 conjugacy classes, 32 normal (30 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6, C8 [×2], C2×C4, D4 [×2], C23 [×2], D5 [×2], C10, C10, C12 [×2], D6 [×4], C2×C6, C15, M4(2) [×2], C2×D4, C20 [×2], D10 [×4], C2×C10, C3⋊C8, C24, D12 [×2], C2×C12, C22×S3 [×2], D15 [×2], C30, C30, C4.D4, C52C8, C40, D20 [×2], C2×C20, C22×D5 [×2], C4.Dic3, C3×M4(2), C2×D12, C60 [×2], D30 [×4], C2×C30, C4.Dic5, C5×M4(2), C2×D20, C12.46D4, C5×C3⋊C8, C3×C52C8, D60 [×2], C2×C60, C22×D15 [×2], C20.46D4, C3×C4.Dic5, C5×C4.Dic3, C2×D60, C60.29D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, C4.D4, C4×D5, D20, C5⋊D4, D6⋊C4, S3×D5, D10⋊C4, C12.46D4, D30.C2, C3⋊D20, C5⋊D12, C20.46D4, D304C4, C60.29D4

Smallest permutation representation of C60.29D4
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)
(1 111 46 66 31 81 16 96)(2 70 47 85 32 100 17 115)(3 89 48 104 33 119 18 74)(4 108 49 63 34 78 19 93)(5 67 50 82 35 97 20 112)(6 86 51 101 36 116 21 71)(7 105 52 120 37 75 22 90)(8 64 53 79 38 94 23 109)(9 83 54 98 39 113 24 68)(10 102 55 117 40 72 25 87)(11 61 56 76 41 91 26 106)(12 80 57 95 42 110 27 65)(13 99 58 114 43 69 28 84)(14 118 59 73 44 88 29 103)(15 77 60 92 45 107 30 62)
(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 45)(18 44)(19 43)(20 42)(21 41)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(61 71)(62 70)(63 69)(64 68)(65 67)(72 120)(73 119)(74 118)(75 117)(76 116)(77 115)(78 114)(79 113)(80 112)(81 111)(82 110)(83 109)(84 108)(85 107)(86 106)(87 105)(88 104)(89 103)(90 102)(91 101)(92 100)(93 99)(94 98)(95 97)```

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

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

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

57 conjugacy classes

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

57 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D5 D6 D10 D12 C3⋊D4 C4×S3 D20 C5⋊D4 C4×D5 C4.D4 S3×D5 C12.46D4 C3⋊D20 C5⋊D12 D30.C2 C20.46D4 C60.29D4 kernel C60.29D4 C3×C4.Dic5 C5×C4.Dic3 C2×D60 C22×D15 C4.Dic5 C60 C4.Dic3 C2×C20 C2×C12 C20 C20 C2×C10 C12 C12 C2×C6 C15 C2×C4 C5 C4 C4 C22 C3 C1 # reps 1 1 1 1 4 1 2 2 1 2 2 2 2 4 4 4 1 2 2 2 2 2 4 8

Matrix representation of C60.29D4 in GL4(𝔽241) generated by

 57 220 0 0 21 191 0 0 0 0 21 184 0 0 57 6
,
 0 0 44 3 0 0 78 197 1 0 0 0 51 240 0 0
,
 1 0 0 0 51 240 0 0 0 0 240 0 0 0 190 1
`G:=sub<GL(4,GF(241))| [57,21,0,0,220,191,0,0,0,0,21,57,0,0,184,6],[0,0,1,51,0,0,0,240,44,78,0,0,3,197,0,0],[1,51,0,0,0,240,0,0,0,0,240,190,0,0,0,1] >;`

C60.29D4 in GAP, Magma, Sage, TeX

`C_{60}._{29}D_4`
`% in TeX`

`G:=Group("C60.29D4");`
`// GroupNames label`

`G:=SmallGroup(480,36);`
`// by ID`

`G=gap.SmallGroup(480,36);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,92,219,100,675,346,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=c^2=1,b^4=a^30,b*a*b^-1=a^19,c*a*c=a^-1,c*b*c=a^45*b^3>;`
`// generators/relations`

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