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## G = C60.38D4order 480 = 25·3·5

### 38th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C60.38D4
 Chief series C1 — C5 — C15 — C30 — C60 — C3×C5⋊2C8 — C5⋊D24 — C60.38D4
 Lower central C15 — C30 — C60 — C60.38D4
 Upper central C1 — C2 — C2×C4

Generators and relations for C60.38D4
G = < a,b,c | a60=c2=1, b4=a30, bab-1=a19, cac=a-1, cbc=a30b3 >

Subgroups: 956 in 136 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, S3 [×3], C6, C6, C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, D5 [×2], C10, C10 [×2], Dic3, C12 [×2], D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C20 [×2], C20, D10 [×4], C2×C10, C2×C10, C24 [×2], Dic6, C4×S3, D12, D12 [×3], C3⋊D4, C2×C12, C22×S3, C5×S3, D15 [×2], C30, C30, C8⋊C22, C52C8 [×2], D20 [×3], C2×C20, C2×C20, C5×D4 [×2], C5×Q8, C22×D5, C24⋊C2 [×2], D24 [×2], C3×M4(2), C2×D12, C4○D12, C5×Dic3, C60 [×2], S3×C10, D30 [×4], C2×C30, C4.Dic5, D4⋊D5 [×2], Q8⋊D5 [×2], C2×D20, C5×C4○D4, C8⋊D6, C3×C52C8 [×2], C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, D60 [×2], D60, C2×C60, C22×D15, D4⋊D10, C5⋊D24 [×2], Dic6⋊D5 [×2], C3×C4.Dic5, C5×C4○D12, C2×D60, C60.38D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, C8⋊C22, C5⋊D4 [×2], C22×D5, C2×D12, S3×D5, C2×C5⋊D4, C8⋊D6, C5⋊D12 [×2], C2×S3×D5, D4⋊D10, C2×C5⋊D12, C60.38D4

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

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

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

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

57 conjugacy classes

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

57 irreducible representations

 dim 1 1 1 1 1 1 2 2 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 C2 C2 S3 D4 D4 D5 D6 D6 D10 D10 D10 D12 D12 C5⋊D4 C5⋊D4 C8⋊C22 S3×D5 C8⋊D6 C5⋊D12 C2×S3×D5 C5⋊D12 D4⋊D10 C60.38D4 kernel C60.38D4 C5⋊D24 Dic6⋊D5 C3×C4.Dic5 C5×C4○D12 C2×D60 C4.Dic5 C60 C2×C30 C4○D12 C5⋊2C8 C2×C20 Dic6 D12 C2×C12 C20 C2×C10 C12 C2×C6 C15 C2×C4 C5 C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 1 1 1 2 2 1 2 2 2 2 2 4 4 1 2 2 2 2 2 4 8

Matrix representation of C60.38D4 in GL4(𝔽241) generated by

 184 21 0 0 220 50 0 0 22 22 220 57 219 180 184 235
,
 214 157 153 235 226 27 85 88 16 212 27 84 64 225 15 214
,
 1 0 0 0 51 240 0 0 120 42 44 3 137 121 78 197
`G:=sub<GL(4,GF(241))| [184,220,22,219,21,50,22,180,0,0,220,184,0,0,57,235],[214,226,16,64,157,27,212,225,153,85,27,15,235,88,84,214],[1,51,120,137,0,240,42,121,0,0,44,78,0,0,3,197] >;`

C60.38D4 in GAP, Magma, Sage, TeX

`C_{60}._{38}D_4`
`% in TeX`

`G:=Group("C60.38D4");`
`// GroupNames label`

`G:=SmallGroup(480,381);`
`// by ID`

`G=gap.SmallGroup(480,381);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,422,100,346,80,1356,18822]);`
`// 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^30*b^3>;`
`// generators/relations`

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