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## G = Dic5×S4order 480 = 25·3·5

### Direct product of Dic5 and S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×A4 — Dic5×S4
 Chief series C1 — C22 — C2×C10 — C5×A4 — C10×A4 — A4×Dic5 — Dic5×S4
 Lower central C5×A4 — Dic5×S4
 Upper central C1 — C2

Generators and relations for Dic5×S4
G = < a,b,c,d,e,f | a10=c2=d2=e3=f2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 640 in 112 conjugacy classes, 23 normal (19 characteristic)
C1, C2, C2 [×4], C3, C4 [×6], C22, C22 [×6], C5, S3 [×2], C6, C2×C4 [×7], D4 [×4], C23, C23, C10, C10 [×4], Dic3, C12, A4, D6, C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5, Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×6], C4×S3, S4 [×2], C2×A4, C5×S3 [×2], C30, C4×D4, C2×Dic5 [×6], C2×C20, C5×D4 [×4], C22×C10, C22×C10, A4⋊C4, C4×A4, C2×S4, C3×Dic5, Dic15, C5×A4, S3×C10, C4×Dic5, C4⋊Dic5, C23.D5 [×2], C22×Dic5, C22×Dic5, D4×C10, C4×S4, S3×Dic5, C5×S4 [×2], C10×A4, D4×Dic5, A4⋊Dic5, A4×Dic5, C10×S4, Dic5×S4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D5, D6, Dic5 [×2], D10, C4×S3, S4, C2×Dic5, C2×S4, S3×D5, C4×S4, S3×Dic5, D5×S4, Dic5×S4

Smallest permutation representation of Dic5×S4
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 36 6 31)(2 35 7 40)(3 34 8 39)(4 33 9 38)(5 32 10 37)(11 44 16 49)(12 43 17 48)(13 42 18 47)(14 41 19 46)(15 50 20 45)(21 54 26 59)(22 53 27 58)(23 52 28 57)(24 51 29 56)(25 60 30 55)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)
(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)
(1 19 29)(2 20 30)(3 11 21)(4 12 22)(5 13 23)(6 14 24)(7 15 25)(8 16 26)(9 17 27)(10 18 28)(31 41 51)(32 42 52)(33 43 53)(34 44 54)(35 45 55)(36 46 56)(37 47 57)(38 48 58)(39 49 59)(40 50 60)
(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 6 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 12A 12B 15A 15B 20A 20B 20C 20D 30A 30B order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 5 5 6 10 10 10 10 10 10 10 10 10 10 12 12 15 15 20 20 20 20 30 30 size 1 1 3 3 6 6 8 5 5 6 6 15 15 30 30 30 30 2 2 8 2 2 6 6 6 6 12 12 12 12 40 40 16 16 12 12 12 12 16 16

40 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 3 3 3 4 4 6 6 type + + + + + + + - + + + + - + - image C1 C2 C2 C2 C4 S3 D5 D6 Dic5 D10 C4×S3 S4 C2×S4 C4×S4 S3×D5 S3×Dic5 D5×S4 Dic5×S4 kernel Dic5×S4 A4⋊Dic5 A4×Dic5 C10×S4 C5×S4 C22×Dic5 C2×S4 C22×C10 S4 C2×A4 C2×C10 Dic5 C10 C5 C23 C22 C2 C1 # reps 1 1 1 1 4 1 2 1 4 2 2 2 2 4 2 2 4 4

Matrix representation of Dic5×S4 in GL5(𝔽61)

 1 60 0 0 0 45 17 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 15 0 0 0 4 0 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 60 0 0 1 0 0
,
 60 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 60

G:=sub<GL(5,GF(61))| [1,45,0,0,0,60,17,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,4,0,0,0,15,0,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,0,0,0,0,0,60,0],[60,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,60] >;

Dic5×S4 in GAP, Magma, Sage, TeX

{\rm Dic}_5\times S_4
% in TeX

G:=Group("Dic5xS4");
// GroupNames label

G:=SmallGroup(480,976);
// by ID

G=gap.SmallGroup(480,976);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,28,234,3364,5052,1286,2953,2232]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^10=c^2=d^2=e^3=f^2=1,b^2=a^5,b*a*b^-1=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,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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