A4xDic5 - GroupNames

G = A₄×Dic₅ order 240 = 2⁴·3·5

Direct product of A₄ and Dic₅

direct product, metabelian, soluble, monomial, A-group

Aliases: A₄×Dic₅, C₅⋊₂(C₄×A₄), (C₅×A₄)⋊₄C₄, C₂.₁(D₅×A₄), (C₂×C₁₀)⋊₂C₁₂, C₂³.(C₃×D₅), C₁₀.₃(C₂×A₄), (C₂×A₄).₂D₅, (C₂²×C₁₀).C₆, C₂²⋊(C₃×Dic₅), (C₂²×Dic₅)⋊C₃, (C₁₀×A₄).₂C₂, SmallGroup(240,110)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — A₄×Dic₅

Chief series

C₁ — C₅ — C₂×C₁₀ — C₂²×C₁₀ — C₁₀×A₄ — A₄×Dic₅

Lower central

C₂×C₁₀ — A₄×Dic₅

Upper central

C₁ — C₂

Generators and relations for A₄×Dic₅
G = < a,b,c,d,e | a²=b²=c³=d¹⁰=1, e²=d⁵, cac^-1=ab=ba, ad=da, ae=ea, cbc^-1=a, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

C₁

C₂

₃C₂

₄C₃

C₅

C₂²

₃C₂²

₅C₄

₁₅C₄

₄C₆

C₁₀

₃C₁₀

₄C₁₅

C₂³

₁₅C₂×C₄

A₄

₂₀C₁₂

C₂×C₁₀

Dic₅

₃C₂×C₁₀

₃Dic₅

₃C₂×C₁₀

₄C₃₀

₅C₂²×C₄

C₂×A₄

C₂²×C₁₀

₃C₂×Dic₅

C₅×A₄

₄C₃×Dic₅

₅C₄×A₄

C₂²×Dic₅

C₁₀×A₄

A₄×Dic₅

Smallest permutation representation of A₄×Dic₅
►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 29 19)(2 30 20)(3 21 11)(4 22 12)(5 23 13)(6 24 14)(7 25 15)(8 26 16)(9 27 17)(10 28 18)(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 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)

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), (1,29,19)(2,30,20)(3,21,11)(4,22,12)(5,23,13)(6,24,14)(7,25,15)(8,26,16)(9,27,17)(10,28,18)(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), (1,36,6,31)(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)>;

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), (1,29,19)(2,30,20)(3,21,11)(4,22,12)(5,23,13)(6,24,14)(7,25,15)(8,26,16)(9,27,17)(10,28,18)(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), (1,36,6,31)(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) );

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)], [(1,29,19),(2,30,20),(3,21,11),(4,22,12),(5,23,13),(6,24,14),(7,25,15),(8,26,16),(9,27,17),(10,28,18),(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)], [(1,36,6,31),(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)]])

A₄×Dic₅ is a maximal subgroup of Dic₅.S₄ A₄⋊Dic₁₀ Dic₅⋊₂S₄ Dic₅⋊S₄ C₄×D₅×A₄
A₄×Dic₅ is a maximal quotient of SL₂(𝔽₃).Dic₅

32 conjugacy classes

class	1	2A	2B	2C	3A	3B	4A	4B	4C	4D	5A	5B	6A	6B	10A	10B	10C	10D	10E	10F	12A	12B	12C	12D	15A	15B	15C	15D	30A	30B	30C	30D
order	1	2	2	2	3	3	4	4	4	4	5	5	6	6	10	10	10	10	10	10	12	12	12	12	15	15	15	15	30	30	30	30
size	1	1	3	3	4	4	5	5	15	15	2	2	4	4	2	2	6	6	6	6	20	20	20	20	8	8	8	8	8	8	8	8

32 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	3	3	3	6	6
type	+	+					+	-			+	+		+	-
image	C₁	C₂	C₃	C₄	C₆	C₁₂	D₅	Dic₅	C₃×D₅	C₃×Dic₅	A₄	C₂×A₄	C₄×A₄	D₅×A₄	A₄×Dic₅
kernel	A₄×Dic₅	C₁₀×A₄	C₂²×Dic₅	C₅×A₄	C₂²×C₁₀	C₂×C₁₀	C₂×A₄	A₄	C₂³	C₂²	Dic₅	C₁₀	C₅	C₂	C₁
# reps	1	1	2	2	2	4	2	2	4	4	1	1	2	2	2

Matrix representation of A₄×Dic₅ ►in GL₅(𝔽₆₁)

1	0	0	0	0
0	1	0	0	0
0	0	60	0	0
0	0	23	1	0
0	0	48	36	60

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	38	60	0
0	0	48	0	60

1	0	0	0	0
0	1	0	0	0
0	0	47	20	0
0	0	56	28	6
0	0	0	0	47

0	60	0	0	0
1	18	0	0	0
0	0	60	0	0
0	0	0	60	0
0	0	0	0	60

8	53	0	0	0
31	53	0	0	0
0	0	11	0	0
0	0	0	11	0
0	0	0	0	11

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,60,23,48,0,0,0,1,36,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,1,38,48,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,47,56,0,0,0,20,28,0,0,0,0,6,47],[0,1,0,0,0,60,18,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[8,31,0,0,0,53,53,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11] >;

A₄×Dic₅ in GAP, Magma, Sage, TeX

A_4\times {\rm Dic}_5

% in TeX

G:=Group("A4xDic5");

// GroupNames label

G:=SmallGroup(240,110);

// by ID

G=gap.SmallGroup(240,110);

# by ID

G:=PCGroup([6,-2,-3,-2,-2,2,-5,36,441,190,6917]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of A₄×Dic₅ in TeX

G = A4×Dic5 order 240 = 24·3·5

Direct product of A4 and Dic5

G = A₄×Dic₅ order 240 = 2⁴·3·5

Direct product of A₄ and Dic₅