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G = S3×D4×D5  order 480 = 25·3·5

Direct product of S3, D4 and D5

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

Aliases: S3×D4×D5, C60⋊C23, D20⋊11D6, D12⋊11D10, D60⋊7C22, D30⋊3C23, C30.25C24, Dic15⋊1C23, (C5×D4)⋊9D6, (C4×D5)⋊7D6, C5⋊D4⋊1D6, (C2×C30)⋊C23, (D4×D15)⋊4C2, (C3×D4)⋊9D10, D15⋊2(C2×D4), (C4×S3)⋊7D10, (S3×D20)⋊4C2, (D5×D12)⋊4C2, C3⋊D4⋊1D10, C15⋊4(C22×D4), C20⋊D6⋊4C2, C20⋊1(C22×S3), (C6×D5)⋊3C23, D6⋊3(C22×D5), C12⋊1(C22×D5), D10⋊D6⋊2C2, (S3×C10)⋊3C23, (S3×C20)⋊1C22, (C3×D20)⋊7C22, (D5×C12)⋊1C22, (C5×D12)⋊7C22, D10⋊3(C22×S3), (D4×C15)⋊7C22, (C4×D15)⋊1C22, (C22×D5)⋊11D6, C5⋊D12⋊3C22, C3⋊D20⋊3C22, C15⋊D4⋊3C22, C15⋊7D4⋊1C22, C6.25(C23×D5), (C22×S3)⋊10D10, C10.25(S3×C23), D30.C2⋊9C22, (S3×Dic5)⋊9C22, Dic3⋊1(C22×D5), (C3×Dic5)⋊1C23, (D5×Dic3)⋊9C22, Dic5⋊1(C22×S3), (C5×Dic3)⋊1C23, (C22×D15)⋊11C22, C3⋊3(C2×D4×D5), C5⋊3(C2×S3×D4), C4⋊1(C2×S3×D5), (C3×D4×D5)⋊4C2, (C4×S3×D5)⋊1C2, (C5×S3×D4)⋊4C2, C22⋊2(C2×S3×D5), (C3×D5)⋊2(C2×D4), (C5×S3)⋊2(C2×D4), (D5×C3⋊D4)⋊2C2, (S3×C5⋊D4)⋊2C2, (D5×C2×C6)⋊7C22, (C22×S3×D5)⋊8C2, (S3×C2×C10)⋊7C22, (C2×S3×D5)⋊12C22, (C2×C6)⋊1(C22×D5), C2.28(C22×S3×D5), (C2×C10)⋊4(C22×S3), (C5×C3⋊D4)⋊1C22, (C3×C5⋊D4)⋊1C22, SmallGroup(480,1097)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — S3×D4×D5
Chief series
C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — C22×S3×D5 — S3×D4×D5
Lower central
C15 — C30 — S3×D4×D5
Upper central
C1 — C2 — D4

Generators and relations for S3×D4×D5
 G = < a,b,c,d,e,f | a3=b2=c4=d2=e5=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 2892 in 472 conjugacy classes, 122 normal (50 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, S3, C6, C6, C2×C4, D4, D4, C23, D5, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C24, Dic5, Dic5, C20, C20, D10, D10, D10, C2×C10, C2×C10, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×S3, C22×C6, C5×S3, C5×S3, C3×D5, C3×D5, D15, D15, C30, C30, C22×D4, C4×D5, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C5×D4, C5×D4, C22×D5, C22×D5, C22×C10, S3×C2×C4, C2×D12, S3×D4, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, S3×D5, C6×D5, C6×D5, C6×D5, S3×C10, S3×C10, S3×C10, D30, D30, D30, C2×C30, C2×C4×D5, C2×D20, D4×D5, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, C2×S3×D4, D5×Dic3, S3×Dic5, D30.C2, C15⋊D4, C3⋊D20, C5⋊D12, D5×C12, C3×D20, C3×C5⋊D4, S3×C20, C5×D12, C5×C3⋊D4, C4×D15, D60, C15⋊7D4, D4×C15, C2×S3×D5, C2×S3×D5, C2×S3×D5, D5×C2×C6, S3×C2×C10, C22×D15, C2×D4×D5, C4×S3×D5, D5×D12, S3×D20, C20⋊D6, D5×C3⋊D4, S3×C5⋊D4, D10⋊D6, C3×D4×D5, C5×S3×D4, D4×D15, C22×S3×D5, S3×D4×D5
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C24, D10, C22×S3, C22×D4, C22×D5, S3×D4, S3×C23, S3×D5, D4×D5, C23×D5, C2×S3×D4, C2×S3×D5, C2×D4×D5, C22×S3×D5, S3×D4×D5

Smallest permutation representation of S3×D4×D5
►On 60 points
Generators in S60
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)

(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)(36 41)(37 42)(38 43)(39 44)(40 45)(51 56)(52 57)(53 58)(54 59)(55 60)

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O 3 4A4B4C4D5A5B6A6B6C6D6E6F6G10A10B10C10D10E10F10G10H10I10J10K10L10M10N12A12B15A15B20A20B20C20D30A30B30C30D30E30F60A60B
order122222222222222234444556666666101010101010101010101010101012121515202020203030303030306060
size1122335566101015153030226103022244101020202244446666121212124204444121244888888

60 irreducible representations

dim1111111111112222222222222444448
type+++++++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D4D5D6D6D6D6D6D10D10D10D10D10S3×D4S3×D5D4×D5C2×S3×D5C2×S3×D5S3×D4×D5
kernelS3×D4×D5C4×S3×D5D5×D12S3×D20C20⋊D6D5×C3⋊D4S3×C5⋊D4D10⋊D6C3×D4×D5C5×S3×D4D4×D15C22×S3×D5D4×D5S3×D5S3×D4C4×D5D20C5⋊D4C5×D4C22×D5C4×S3D12C3⋊D4C3×D4C22×S3D5D4S3C4C22C1
# reps1111122211121421121222424224242

Matrix representation of S3×D4×D5 ►in GL6(𝔽61)

0600000
1600000
001000
000100
000010
000001
,
010000
100000
0060000
0006000
000010
000001
,
6000000
0600000
0060000
0006000
0000141
00005560
,
100000
010000
001000
000100
000010
00005560
,
100000
010000
0044100
00166000
000010
000001
,
100000
010000
00606000
000100
000010
000001

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,55,0,0,0,0,41,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,55,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,44,16,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

S3×D4×D5 in GAP, Magma, Sage, TeX 

S_3\times D_4\times D_5
 
% in TeX
 
G:=Group("S3xD4xD5");
 
// GroupNames label
 
G:=SmallGroup(480,1097);
 
// by ID
 
G=gap.SmallGroup(480,1097);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,185,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^2=c^4=d^2=e^5=f^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
 
// generators/relations
 

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