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## G = S3×D4×D5order 480 = 25·3·5

### Direct product of S3, D4 and D5

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, C157D4, 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 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 10M 10N 12A 12B 15A 15B 20A 20B 20C 20D 30A 30B 30C 30D 30E 30F 60A 60B order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 6 6 6 6 10 10 10 10 10 10 10 10 10 10 10 10 10 10 12 12 15 15 20 20 20 20 30 30 30 30 30 30 60 60 size 1 1 2 2 3 3 5 5 6 6 10 10 15 15 30 30 2 2 6 10 30 2 2 2 4 4 10 10 20 20 2 2 4 4 4 4 6 6 6 6 12 12 12 12 4 20 4 4 4 4 12 12 4 4 8 8 8 8 8 8

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D5 D6 D6 D6 D6 D6 D10 D10 D10 D10 D10 S3×D4 S3×D5 D4×D5 C2×S3×D5 C2×S3×D5 S3×D4×D5 kernel S3×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 D4×D5 S3×D5 S3×D4 C4×D5 D20 C5⋊D4 C5×D4 C22×D5 C4×S3 D12 C3⋊D4 C3×D4 C22×S3 D5 D4 S3 C4 C22 C1 # reps 1 1 1 1 1 2 2 2 1 1 1 2 1 4 2 1 1 2 1 2 2 2 4 2 4 2 2 4 2 4 2

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

 0 60 0 0 0 0 1 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 41 0 0 0 0 55 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 0 0 0 0 0 55 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 44 1 0 0 0 0 16 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 60 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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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