Dic5:2S4 - GroupNames

G = Dic₅⋊₂S₄ order 480 = 2⁵·3·5

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

Aliases: Dic₅⋊₂S₄, C₅⋊S₄⋊₂C₄, C₅⋊₂(C₄×S₄), A₄⋊C₄⋊₂D₅, A₄⋊₁(C₄×D₅), C₂.₂(D₅×S₄), C₁₀.₁₁(C₂×S₄), (A₄×Dic₅)⋊₂C₂, (C₂×A₄).₃D₁₀, C₂³.₃(S₃×D₅), C₂²⋊(D₃₀.C₂), (C₂²×C₁₀).₃D₆, (C₁₀×A₄).₃C₂², (C₂²×Dic₅)⋊₂S₃, (C₂×C₅⋊S₄).C₂, (C₅×A₄⋊C₄)⋊₁C₂, (C₅×A₄)⋊₅(C₂×C₄), (C₂×C₁₀)⋊₃(C₄×S₃), SmallGroup(480,977)

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 Dic₅⋊₂S₄
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, C₃, C₄, C₂², C₂², C₅, S₃, C₆, C₂×C₄, D₄, C₂³, C₂³, D₅, C₁₀, C₁₀, Dic₃, C₁₂, A₄, D₆, C₁₅, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, Dic₅, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₄×S₃, S₄, C₂×A₄, D₁₅, C₃₀, C₄×D₄, C₄×D₅, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, 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, C₁₀×A₄, Dic₅⋊₄D₄, C₅×A₄⋊C₄, A₄×Dic₅, C₂×C₅⋊S₄, Dic₅⋊₂S₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₅, D₆, D₁₀, C₄×S₃, S₄, C₄×D₅, C₂×S₄, S₃×D₅, C₄×S₄, D₃₀.C₂, D₅×S₄, Dic₅⋊₂S₄

Smallest permutation representation of Dic₅⋊₂S₄
►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 Dic₅⋊₂S₄ ►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] >;

Dic₅⋊₂S₄ 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

G = Dic5⋊2S4 order 480 = 25·3·5

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

G = Dic₅⋊₂S₄ order 480 = 2⁵·3·5

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