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G = D60⋊3C4  order 480 = 25·3·5

3rd semidirect product of D60 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D60⋊3C4, D12⋊5F5, C3⋊F5⋊1D4, C3⋊2(D4×F5), C15⋊2(C4×D4), C4⋊F5⋊2S3, C4⋊2(S3×F5), C20⋊2(C4×S3), C60⋊1(C2×C4), D6⋊F5⋊2C2, D6⋊2(C2×F5), C12⋊2(C2×F5), C5⋊(Dic3⋊5D4), (C5×D12)⋊3C4, D30⋊2(C2×C4), C5⋊D12⋊2C4, D5.2(S3×D4), (C2×F5).4D6, Dic5⋊2(C4×S3), (C4×D5).38D6, (D5×D12).4C2, (C6×F5).4C22, C6.16(C22×F5), C30.16(C22×C4), (C6×D5).28C23, D5.2(Q8⋊3S3), D10.31(C22×S3), (D5×C12).37C22, (C4×C3⋊F5)⋊3C2, (C2×S3×F5)⋊2C2, (C3×C4⋊F5)⋊2C2, C2.19(C2×S3×F5), C10.16(S3×C2×C4), (S3×C10)⋊2(C2×C4), (C3×D5).3(C2×D4), (C2×S3×D5).2C22, (C2×C3⋊F5).10C22, (C3×D5).6(C4○D4), (C3×Dic5)⋊10(C2×C4), SmallGroup(480,997)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — D60⋊3C4
Chief series
C1 — C5 — C15 — C3×D5 — C6×D5 — C6×F5 — C2×S3×F5 — D60⋊3C4
Lower central
C15 — C30 — D60⋊3C4
Upper central
C1 — C2 — C4

Generators and relations for D60⋊3C4
 G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a43, cbc-1=a42b >

Subgroups: 1172 in 188 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, C23, D5, D5, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, F5, D10, D10, C2×C10, C4×S3, D12, D12, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C4×D4, C4×D5, D20, C5⋊D4, C5×D4, C2×F5, C2×F5, C22×D5, C4×Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C3×Dic5, C60, C3×F5, C3⋊F5, C3⋊F5, S3×D5, C6×D5, S3×C10, D30, C4×F5, C4⋊F5, C22⋊F5, D4×D5, C22×F5, Dic3⋊5D4, C5⋊D12, D5×C12, C5×D12, D60, S3×F5, C6×F5, C2×C3⋊F5, C2×S3×D5, D4×F5, D6⋊F5, C3×C4⋊F5, C4×C3⋊F5, D5×D12, C2×S3×F5, D60⋊3C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, F5, C4×S3, C22×S3, C4×D4, C2×F5, S3×C2×C4, S3×D4, Q8⋊3S3, C22×F5, Dic3⋊5D4, S3×F5, D4×F5, C2×S3×F5, D60⋊3C4

Smallest permutation representation of D60⋊3C4
►On 60 points
Generators in S60
(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)

(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 60)(17 59)(18 58)(19 57)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)

(1 16)(2 23 50 59)(3 30 39 42)(4 37 28 25)(5 44 17 8)(6 51)(7 58 55 34)(9 12 33 60)(10 19 22 43)(11 26)(13 40 49 52)(14 47 38 35)(15 54 27 18)(20 29 32 53)(21 36)(24 57 48 45)(31 46)(41 56)
 
G:=sub<Sym(60)| (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,60)(17,59)(18,58)(19,57)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39), (1,16)(2,23,50,59)(3,30,39,42)(4,37,28,25)(5,44,17,8)(6,51)(7,58,55,34)(9,12,33,60)(10,19,22,43)(11,26)(13,40,49,52)(14,47,38,35)(15,54,27,18)(20,29,32,53)(21,36)(24,57,48,45)(31,46)(41,56)>;
 
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,60)(17,59)(18,58)(19,57)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39), (1,16)(2,23,50,59)(3,30,39,42)(4,37,28,25)(5,44,17,8)(6,51)(7,58,55,34)(9,12,33,60)(10,19,22,43)(11,26)(13,40,49,52)(14,47,38,35)(15,54,27,18)(20,29,32,53)(21,36)(24,57,48,45)(31,46)(41,56) );
 
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,60),(17,59),(18,58),(19,57),(20,56),(21,55),(22,54),(23,53),(24,52),(25,51),(26,50),(27,49),(28,48),(29,47),(30,46),(31,45),(32,44),(33,43),(34,42),(35,41),(36,40),(37,39)], [(1,16),(2,23,50,59),(3,30,39,42),(4,37,28,25),(5,44,17,8),(6,51),(7,58,55,34),(9,12,33,60),(10,19,22,43),(11,26),(13,40,49,52),(14,47,38,35),(15,54,27,18),(20,29,32,53),(21,36),(24,57,48,45),(31,46),(41,56)]])
 

39 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B···4F4G4H4I4J4K4L 5 6A6B6C10A10B10C12A12B···12F 15  20  30 60A60B
order12222222344···444444456661010101212···121520306060
size11556630302210···1015151515303042101042424420···2088888

39 irreducible representations

dim1111111112222222444448888
type+++++++++++++++++++
imageC1C2C2C2C2C2C4C4C4S3D4D6D6C4○D4C4×S3C4×S3F5C2×F5C2×F5S3×D4Q8⋊3S3S3×F5D4×F5C2×S3×F5D60⋊3C4
kernelD60⋊3C4D6⋊F5C3×C4⋊F5C4×C3⋊F5D5×D12C2×S3×F5C5⋊D12C5×D12D60C4⋊F5C3⋊F5C4×D5C2×F5C3×D5Dic5C20D12C12D6D5D5C4C3C2C1
# reps1211124221212222112111112

Matrix representation of D60⋊3C4 ►in GL8(ℤ)

000-10000
001-10000
01000000
-11000000
00000010
00000001
0000-1-1-1-1
00001000
,
-11000000
01000000
001-10000
000-10000
00000010
00000100
00001000
0000-1-1-1-1
,
00100000
00010000
10000000
01000000
00001000
00000001
00000100
0000-1-1-1-1

G:=sub<GL(8,Integers())| [0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0],[-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,1,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,-1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1] >;
 

D60⋊3C4 in GAP, Magma, Sage, TeX 

D_{60}\rtimes_3C_4
 
% in TeX
 
G:=Group("D60:3C4");
 
// GroupNames label
 
G:=SmallGroup(480,997);
 
// by ID
 
G=gap.SmallGroup(480,997);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,219,100,1356,9414,2379]);
 
// Polycyclic
 
G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^43,c*b*c^-1=a^42*b>;
 
// generators/relations
 

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