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## G = C5×S3×D4order 240 = 24·3·5

### Direct product of C5, S3 and D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×S3×D4
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C2×C10 — C5×S3×D4
 Lower central C3 — C6 — C5×S3×D4
 Upper central C1 — C10 — C5×D4

Generators and relations for C5×S3×D4
G = < a,b,c,d,e | a5=b3=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 240 in 108 conjugacy classes, 50 normal (26 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22 [×2], C22 [×7], C5, S3 [×2], S3 [×2], C6, C6 [×2], C2×C4, D4, D4 [×3], C23 [×2], C10, C10 [×6], Dic3, C12, D6, D6 [×2], D6 [×4], C2×C6 [×2], C15, C2×D4, C20, C20, C2×C10 [×2], C2×C10 [×7], C4×S3, D12, C3⋊D4 [×2], C3×D4, C22×S3 [×2], C5×S3 [×2], C5×S3 [×2], C30, C30 [×2], C2×C20, C5×D4, C5×D4 [×3], C22×C10 [×2], S3×D4, C5×Dic3, C60, S3×C10, S3×C10 [×2], S3×C10 [×4], C2×C30 [×2], D4×C10, S3×C20, C5×D12, C5×C3⋊D4 [×2], D4×C15, S3×C2×C10 [×2], C5×S3×D4
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], C22×S3, C5×S3, C5×D4 [×2], C22×C10, S3×D4, S3×C10 [×3], D4×C10, S3×C2×C10, C5×S3×D4

Smallest permutation representation of C5×S3×D4
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 45 33)(2 41 34)(3 42 35)(4 43 31)(5 44 32)(6 17 15)(7 18 11)(8 19 12)(9 20 13)(10 16 14)(21 58 51)(22 59 52)(23 60 53)(24 56 54)(25 57 55)(26 47 40)(27 48 36)(28 49 37)(29 50 38)(30 46 39)
(11 18)(12 19)(13 20)(14 16)(15 17)(21 51)(22 52)(23 53)(24 54)(25 55)(31 43)(32 44)(33 45)(34 41)(35 42)(36 48)(37 49)(38 50)(39 46)(40 47)
(1 6 26 56)(2 7 27 57)(3 8 28 58)(4 9 29 59)(5 10 30 60)(11 36 25 34)(12 37 21 35)(13 38 22 31)(14 39 23 32)(15 40 24 33)(16 46 53 44)(17 47 54 45)(18 48 55 41)(19 49 51 42)(20 50 52 43)
(6 56)(7 57)(8 58)(9 59)(10 60)(11 25)(12 21)(13 22)(14 23)(15 24)(16 53)(17 54)(18 55)(19 51)(20 52)

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

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

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

C5×S3×D4 is a maximal subgroup of   D2010D6  D12.9D10  D2013D6  D1214D10

75 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 5C 5D 6A 6B 6C 10A 10B 10C 10D 10E ··· 10L 10M ··· 10T 10U ··· 10AB 12 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 20G 20H 30A 30B 30C 30D 30E ··· 30L 60A 60B 60C 60D order 1 2 2 2 2 2 2 2 3 4 4 5 5 5 5 6 6 6 10 10 10 10 10 ··· 10 10 ··· 10 10 ··· 10 12 15 15 15 15 20 20 20 20 20 20 20 20 30 30 30 30 30 ··· 30 60 60 60 60 size 1 1 2 2 3 3 6 6 2 2 6 1 1 1 1 2 4 4 1 1 1 1 2 ··· 2 3 ··· 3 6 ··· 6 4 2 2 2 2 2 2 2 2 6 6 6 6 2 2 2 2 4 ··· 4 4 4 4 4

75 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 S3 D4 D6 D6 C5×S3 C5×D4 S3×C10 S3×C10 S3×D4 C5×S3×D4 kernel C5×S3×D4 S3×C20 C5×D12 C5×C3⋊D4 D4×C15 S3×C2×C10 S3×D4 C4×S3 D12 C3⋊D4 C3×D4 C22×S3 C5×D4 C5×S3 C20 C2×C10 D4 S3 C4 C22 C5 C1 # reps 1 1 1 2 1 2 4 4 4 8 4 8 1 2 1 2 4 8 4 8 1 4

Matrix representation of C5×S3×D4 in GL4(𝔽61) generated by

 20 0 0 0 0 20 0 0 0 0 34 0 0 0 0 34
,
 0 60 0 0 1 60 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 60 0 0 0 0 1 2 0 0 60 60
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 60 60
G:=sub<GL(4,GF(61))| [20,0,0,0,0,20,0,0,0,0,34,0,0,0,0,34],[0,1,0,0,60,60,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,1,60,0,0,2,60],[1,0,0,0,0,1,0,0,0,0,1,60,0,0,0,60] >;

C5×S3×D4 in GAP, Magma, Sage, TeX

C_5\times S_3\times D_4
% in TeX

G:=Group("C5xS3xD4");
// GroupNames label

G:=SmallGroup(240,169);
// by ID

G=gap.SmallGroup(240,169);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-3,404,5765]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^3=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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