A4xC5:D4 - GroupNames

G = A₄×C₅⋊D₄ order 480 = 2⁵·3·5

Direct product of A₄ and C₅⋊D₄

direct product, metabelian, soluble, monomial

Aliases: A₄×C₅⋊D₄, C₅⋊₂(D₄×A₄), (C₅×A₄)⋊₈D₄, Dic₅⋊(C₂×A₄), D₁₀⋊₂(C₂×A₄), C₂²⋊₂(D₅×A₄), C₂⁴⋊₄(C₃×D₅), (C₂³×C₁₀)⋊₄C₆, (A₄×Dic₅)⋊₄C₂, (C₂³×D₅)⋊₂C₆, (C₂²×A₄)⋊₁D₅, (C₂²×Dic₅)⋊C₆, (C₂×A₄).₁₇D₁₀, C₂³.₁₄(C₆×D₅), C₁₀.₁₁(C₂²×A₄), (C₁₀×A₄).₁₇C₂², (C₂×D₅×A₄)⋊₅C₂, (A₄×C₂×C₁₀)⋊₄C₂, (C₂²×C₅⋊D₄)⋊C₃, C₂²⋊(C₃×C₅⋊D₄), C₂.₁₁(C₂×D₅×A₄), (C₂×C₁₀)⋊₄(C₂×A₄), (C₂×C₁₀)⋊₂(C₃×D₄), (C₂²×C₁₀).₅(C₂×C₆), SmallGroup(480,1045)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₀ — A₄×C₅⋊D₄

Chief series

C₁ — C₅ — C₂×C₁₀ — C₂²×C₁₀ — C₁₀×A₄ — C₂×D₅×A₄ — A₄×C₅⋊D₄

Lower central

C₂×C₁₀ — C₂²×C₁₀ — A₄×C₅⋊D₄

Upper central

C₁ — C₂ — C₂²

Generators and relations for A₄×C₅⋊D₄
G = < a,b,c,d,e,f | a²=b²=c³=d⁵=e⁴=f²=1, cac^-1=ab=ba, ad=da, ae=ea, af=fa, cbc^-1=a, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede^-1=fdf=d^-1, fef=e^-1 >

Subgroups: 880 in 132 conjugacy classes, 27 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₅, C₆, C₂×C₄, D₄, C₂³, C₂³, D₅, C₁₀, C₁₀, C₁₂, A₄, C₂×C₆, C₁₅, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, Dic₅, Dic₅, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₃×D₄, C₂×A₄, C₂×A₄, C₃×D₅, C₃₀, C₂²×D₄, C₂×Dic₅, C₅⋊D₄, C₅⋊D₄, C₂²×D₅, C₂²×C₁₀, C₂²×C₁₀, C₄×A₄, C₂²×A₄, C₂²×A₄, C₃×Dic₅, C₅×A₄, C₆×D₅, C₂×C₃₀, C₂²×Dic₅, C₂×C₅⋊D₄, C₂³×D₅, C₂³×C₁₀, D₄×A₄, C₃×C₅⋊D₄, D₅×A₄, C₁₀×A₄, C₁₀×A₄, C₂²×C₅⋊D₄, A₄×Dic₅, C₂×D₅×A₄, A₄×C₂×C₁₀, A₄×C₅⋊D₄
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, D₅, A₄, C₂×C₆, D₁₀, C₃×D₄, C₂×A₄, C₃×D₅, C₅⋊D₄, C₂²×A₄, C₆×D₅, D₄×A₄, C₃×C₅⋊D₄, D₅×A₄, C₂×D₅×A₄, A₄×C₅⋊D₄

Smallest permutation representation of A₄×C₅⋊D₄
►On 60 points

Generators in S₆₀

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

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

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

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

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

52 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	4A	4B	5A	5B	6A	6B	6C	6D	6E	6F	10A	···	10F	10G	···	10N	12A	12B	15A	15B	15C	15D	30A	···	30L
order	1	2	2	2	2	2	2	2	3	3	4	4	5	5	6	6	6	6	6	6	10	···	10	10	···	10	12	12	15	15	15	15	30	···	30
size	1	1	2	3	3	6	10	30	4	4	10	30	2	2	4	4	8	8	40	40	2	···	2	6	···	6	40	40	8	8	8	8	8	···	8

52 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	3	3	3	3	6	6	6	6
type	+	+	+	+					+	+	+						+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	D₄	D₅	D₁₀	C₃×D₄	C₃×D₅	C₅⋊D₄	C₆×D₅	C₃×C₅⋊D₄	A₄	C₂×A₄	C₂×A₄	C₂×A₄	D₄×A₄	D₅×A₄	C₂×D₅×A₄	A₄×C₅⋊D₄
kernel	A₄×C₅⋊D₄	A₄×Dic₅	C₂×D₅×A₄	A₄×C₂×C₁₀	C₂²×C₅⋊D₄	C₂²×Dic₅	C₂³×D₅	C₂³×C₁₀	C₅×A₄	C₂²×A₄	C₂×A₄	C₂×C₁₀	C₂⁴	A₄	C₂³	C₂²	C₅⋊D₄	Dic₅	D₁₀	C₂×C₁₀	C₅	C₂²	C₂	C₁
# reps	1	1	1	1	2	2	2	2	1	2	2	2	4	4	4	8	1	1	1	1	1	2	2	4

Matrix representation of A₄×C₅⋊D₄ ►in GL₅(𝔽₆₁)

1	0	0	0	0
0	1	0	0	0
0	0	0	60	1
0	0	0	60	0
0	0	1	60	0

1	0	0	0	0
0	1	0	0	0
0	0	60	0	0
0	0	60	0	1
0	0	60	1	0

1	0	0	0	0
0	1	0	0	0
0	0	0	1	0
0	0	0	0	1
0	0	1	0	0

52	18	0	0	0
18	52	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

21	25	0	0	0
36	40	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

36	40	0	0	0
21	25	0	0	0
0	0	60	0	0
0	0	0	60	0
0	0	0	0	60

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,60,60,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,60,60,60,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[52,18,0,0,0,18,52,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[21,36,0,0,0,25,40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[36,21,0,0,0,40,25,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60] >;

A₄×C₅⋊D₄ in GAP, Magma, Sage, TeX

A_4\times C_5\rtimes D_4

% in TeX

G:=Group("A4xC5:D4");

// GroupNames label

G:=SmallGroup(480,1045);

// by ID

G=gap.SmallGroup(480,1045);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-5,197,648,271,18822]);

// Polycyclic

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

// generators/relations

G = A4×C5⋊D4 order 480 = 25·3·5

Direct product of A4 and C5⋊D4

G = A₄×C₅⋊D₄ order 480 = 2⁵·3·5

Direct product of A₄ and C₅⋊D₄