Dic5xS4 - GroupNames

G = Dic₅×S₄ order 480 = 2⁵·3·5

Direct product of Dic₅ and S₄

direct product, non-abelian, soluble, monomial

Aliases: Dic₅×S₄, C₅⋊₄(C₄×S₄), (C₂×S₄).D₅, (C₅×S₄)⋊₂C₄, (C₁₀×S₄).C₂, C₂.₁(D₅×S₄), C₂²⋊(S₃×Dic₅), C₁₀.₁₀(C₂×S₄), A₄⋊Dic₅⋊₂C₂, (A₄×Dic₅)⋊₁C₂, A₄⋊₁(C₂×Dic₅), (C₂×A₄).₂D₁₀, C₂³.₂(S₃×D₅), (C₂²×C₁₀).₂D₆, (C₁₀×A₄).₂C₂², (C₂²×Dic₅)⋊₁S₃, (C₅×A₄)⋊₄(C₂×C₄), (C₂×C₁₀)⋊₂(C₄×S₃), SmallGroup(480,976)

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, C₃, C₄, C₂², C₂², C₅, S₃, C₆, C₂×C₄, D₄, C₂³, C₂³, C₁₀, C₁₀, Dic₃, C₁₂, A₄, D₆, C₁₅, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, Dic₅, Dic₅, C₂₀, C₂×C₁₀, C₂×C₁₀, C₄×S₃, S₄, C₂×A₄, C₅×S₃, C₃₀, C₄×D₄, C₂×Dic₅, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, A₄⋊C₄, C₄×A₄, C₂×S₄, C₃×Dic₅, Dic₁₅, C₅×A₄, S₃×C₁₀, C₄×Dic₅, C₄⋊Dic₅, C₂³.D₅, C₂²×Dic₅, C₂²×Dic₅, D₄×C₁₀, C₄×S₄, S₃×Dic₅, C₅×S₄, C₁₀×A₄, D₄×Dic₅, A₄⋊Dic₅, A₄×Dic₅, C₁₀×S₄, Dic₅×S₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₅, D₆, Dic₅, 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

G = Dic5×S4 order 480 = 25·3·5

Direct product of Dic5 and S4

G = Dic₅×S₄ order 480 = 2⁵·3·5

Direct product of Dic₅ and S₄