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G = F5×D12  order 480 = 25·3·5

Direct product of F5 and D12

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: F5×D12, D60⋊6C4, C5⋊(C4×D12), C3⋊1(D4×F5), C15⋊1(C4×D4), C4⋊1(S3×F5), C20⋊1(C4×S3), C60⋊2(C2×C4), D6⋊F5⋊1C2, (C3×F5)⋊1D4, (C4×F5)⋊3S3, D6⋊1(C2×F5), C12⋊4(C2×F5), D30⋊1(C2×C4), (C5×D12)⋊6C4, C5⋊D12⋊1C4, (C12×F5)⋊3C2, C60⋊C4⋊2C2, Dic5⋊1(C4×S3), (C4×D5).37D6, (D5×D12).6C2, D5.1(C2×D12), (C2×F5).11D6, C6.14(C22×F5), D5.3(C4○D12), C30.14(C22×C4), (C6×D5).26C23, (C6×F5).10C22, D10.29(C22×S3), (D5×C12).45C22, (C2×S3×F5)⋊1C2, C2.17(C2×S3×F5), C10.14(S3×C2×C4), (S3×C10)⋊1(C2×C4), (C3×D5).1(C2×D4), (C2×S3×D5).1C22, (C2×C3⋊F5).3C22, (C3×Dic5)⋊8(C2×C4), (C3×D5).5(C4○D4), SmallGroup(480,995)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — F5×D12
Chief series
C1 — C5 — C15 — C3×D5 — C6×D5 — C6×F5 — C2×S3×F5 — F5×D12
Lower central
C15 — C30 — F5×D12
Upper central
C1 — C2 — C4

Generators and relations for F5×D12
 G = < a,b,c,d | a5=b4=c12=d2=1, bab-1=a3, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1172 in 188 conjugacy classes, 54 normal (34 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, 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, C4×C12, 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, C4×D12, C5⋊D12, D5×C12, C5×D12, D60, S3×F5, C6×F5, C2×C3⋊F5, C2×S3×D5, D4×F5, D6⋊F5, C12×F5, C60⋊C4, D5×D12, C2×S3×F5, F5×D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, F5, C4×S3, D12, C22×S3, C4×D4, C2×F5, S3×C2×C4, C2×D12, C4○D12, C22×F5, C4×D12, S3×F5, D4×F5, C2×S3×F5, F5×D12

Smallest permutation representation of F5×D12
►On 60 points
Generators in S60
(1 30 59 17 47)(2 31 60 18 48)(3 32 49 19 37)(4 33 50 20 38)(5 34 51 21 39)(6 35 52 22 40)(7 36 53 23 41)(8 25 54 24 42)(9 26 55 13 43)(10 27 56 14 44)(11 28 57 15 45)(12 29 58 16 46)

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

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

45 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I4J4K4L 5 6A6B6C10A10B10C12A12B12C···12L 15  20  30 60A60B
order1222222234444444444445666101010121212···121520306060
size115566303022555510101030303030421010424242210···1088888

45 irreducible representations

dim1111111112222222224448888
type++++++++++++++++++
imageC1C2C2C2C2C2C4C4C4S3D4D6D6C4○D4C4×S3C4×S3D12C4○D12F5C2×F5C2×F5S3×F5D4×F5C2×S3×F5F5×D12
kernelF5×D12D6⋊F5C12×F5C60⋊C4D5×D12C2×S3×F5C5⋊D12C5×D12D60C4×F5C3×F5C4×D5C2×F5C3×D5Dic5C20F5D5D12C12D6C4C3C2C1
# reps1211124221212222441121112

Matrix representation of F5×D12 ►in GL8(𝔽61)

10000000
01000000
00100000
00010000
000000060
000010060
000001060
000000160
,
600000000
060000000
005000000
000500000
00000010
00001000
00000001
00000100
,
01000000
600000000
000600000
001600000
000060000
000006000
000000600
000000060
,
600000000
01000000
001600000
000600000
000060000
000006000
000000600
000000060

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,60,60,60],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60] >;
 

F5×D12 in GAP, Magma, Sage, TeX 

F_5\times D_{12}
 
% in TeX
 
G:=Group("F5xD12");
 
// GroupNames label
 
G:=SmallGroup(480,995);
 
// by ID
 
G=gap.SmallGroup(480,995);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,219,100,1356,9414,2379]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^4=c^12=d^2=1,b*a*b^-1=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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