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## G = Dic5⋊2S4order 480 = 25·3·5

### The semidirect product of Dic5 and S4 acting through Inn(Dic5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×A4 — Dic5⋊2S4
 Chief series C1 — C22 — C2×C10 — C5×A4 — C10×A4 — A4×Dic5 — Dic5⋊2S4
 Lower central C5×A4 — Dic5⋊2S4
 Upper central C1 — C2

Generators and relations for Dic52S4
G = < a,b,c,d,e,f | a10=c2=d2=e3=f2=1, b2=a5, bab-1=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 824 in 112 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2 [×4], C3, C4 [×6], C22, C22 [×6], C5, S3 [×2], C6, C2×C4 [×7], D4 [×4], C23, C23, D5 [×2], C10, C10 [×2], Dic3, C12, A4, D6, C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5, Dic5 [×3], C20 [×2], D10 [×4], C2×C10, C2×C10 [×2], C4×S3, S4 [×2], C2×A4, D15 [×2], C30, C4×D4, C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×4], C2×C20 [×2], C22×D5, C22×C10, A4⋊C4, C4×A4, C2×S4, C5×Dic3, C3×Dic5, C5×A4, D30, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C4×S4, D30.C2, C5⋊S4 [×2], C10×A4, Dic54D4, C5×A4⋊C4, A4×Dic5, C2×C5⋊S4, Dic52S4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D5, D6, D10, C4×S3, S4, C4×D5, C2×S4, S3×D5, C4×S4, D30.C2, D5×S4, Dic52S4

Smallest permutation representation of Dic52S4
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 34 6 39)(2 33 7 38)(3 32 8 37)(4 31 9 36)(5 40 10 35)(11 52 16 57)(12 51 17 56)(13 60 18 55)(14 59 19 54)(15 58 20 53)(21 42 26 47)(22 41 27 46)(23 50 28 45)(24 49 29 44)(25 48 30 43)
(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)(51 56)(52 57)(53 58)(54 59)(55 60)
(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 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)
(1 6)(2 5)(3 4)(7 10)(8 9)(11 22)(12 21)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(31 32)(33 40)(34 39)(35 38)(36 37)(41 52)(42 51)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(49 54)(50 53)

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

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

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

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 12A 12B 15A 15B 20A ··· 20H 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 12 12 15 15 20 ··· 20 30 30 size 1 1 3 3 30 30 8 5 5 6 6 6 6 15 15 30 30 2 2 8 2 2 6 6 6 6 40 40 16 16 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 D10 C4×S3 C4×D5 S4 C2×S4 C4×S4 S3×D5 D30.C2 D5×S4 Dic5⋊2S4 kernel Dic5⋊2S4 C5×A4⋊C4 A4×Dic5 C2×C5⋊S4 C5⋊S4 C22×Dic5 A4⋊C4 C22×C10 C2×A4 C2×C10 A4 Dic5 C10 C5 C23 C22 C2 C1 # reps 1 1 1 1 4 1 2 1 2 2 4 2 2 4 2 2 4 4

Matrix representation of Dic52S4 in GL5(𝔽61)

 0 60 0 0 0 1 44 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 50 0 0 0 0 57 11 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 60 60 0 0 0 1 0 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 60 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 2 0 0 0 0 60 60 0 0 0 1 0
,
 60 0 0 0 0 44 1 0 0 0 0 0 60 59 0 0 0 0 1 0 0 0 0 60 60

G:=sub<GL(5,GF(61))| [0,1,0,0,0,60,44,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[50,57,0,0,0,0,11,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,1,60,1,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,60,0,0,0,60,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2,60,1,0,0,0,60,0],[60,44,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,59,1,60,0,0,0,0,60] >;

Dic52S4 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_2S_4
% in TeX

G:=Group("Dic5:2S4");
// GroupNames label

G:=SmallGroup(480,977);
// by ID

G=gap.SmallGroup(480,977);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,28,36,234,3364,5052,1286,2953,2232]);
// 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=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*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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