C2xA4^2 - GroupNames

G = C₂×A₄² order 288 = 2⁵·3²

Direct product of C₂, A₄ and A₄

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

Aliases: C₂×A₄², C₂⁵⋊C₃², C₂⁴⋊(C₃×C₆), (C₂³×A₄)⋊C₃, C₂²⋊A₄⋊₂C₆, C₂²⋊₁(C₆×A₄), C₂³⋊₃(C₃×A₄), (C₂²×A₄)⋊₂C₆, (C₂×C₂²⋊A₄)⋊C₃, SmallGroup(288,1029)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₂×A₄²

Chief series

C₁ — C₂² — C₂⁴ — C₂²×A₄ — A₄² — C₂×A₄²

Lower central

C₂⁴ — C₂×A₄²

Upper central

C₁ — C₂

Generators and relations for C₂×A₄²
G = < a,b,c,d,e,f,g | a²=b²=c²=d³=e²=f²=g³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, dbd^-1=bc=cb, be=eb, bf=fb, bg=gb, dcd^-1=b, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg^-1=ef=fe, gfg^-1=e >

Subgroups: 732 in 122 conjugacy classes, 22 normal (6 characteristic)
C₁, C₂, C₂, C₃, C₂², C₂², C₆, C₂³, C₂³, C₃², A₄, A₄, C₂×C₆, C₂⁴, C₂⁴, C₃×C₆, C₂×A₄, C₂×A₄, C₂²×C₆, C₂⁵, C₃×A₄, C₂²×A₄, C₂²×A₄, C₂²⋊A₄, C₆×A₄, C₂³×A₄, C₂×C₂²⋊A₄, A₄², C₂×A₄²
Quotients: C₁, C₂, C₃, C₆, C₃², A₄, C₃×C₆, C₂×A₄, C₃×A₄, C₆×A₄, A₄², C₂×A₄²

Permutation representations of C₂×A₄²
►On 18 points - transitive group 18T109

Generators in S₁₈

(1 7)(2 8)(3 9)(4 16)(5 17)(6 18)(10 13)(11 14)(12 15)

(1 7)(2 8)(4 16)(5 17)(10 13)(11 14)

(2 8)(3 9)(5 17)(6 18)(11 14)(12 15)

(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)

(1 7)(2 8)(3 9)(4 16)(5 17)(6 18)

(4 16)(5 17)(6 18)(10 13)(11 14)(12 15)

(1 16 10)(2 17 11)(3 18 12)(4 13 7)(5 14 8)(6 15 9)

G:=sub<Sym(18)| (1,7)(2,8)(3,9)(4,16)(5,17)(6,18)(10,13)(11,14)(12,15), (1,7)(2,8)(4,16)(5,17)(10,13)(11,14), (2,8)(3,9)(5,17)(6,18)(11,14)(12,15), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,7)(2,8)(3,9)(4,16)(5,17)(6,18), (4,16)(5,17)(6,18)(10,13)(11,14)(12,15), (1,16,10)(2,17,11)(3,18,12)(4,13,7)(5,14,8)(6,15,9)>;

G:=Group( (1,7)(2,8)(3,9)(4,16)(5,17)(6,18)(10,13)(11,14)(12,15), (1,7)(2,8)(4,16)(5,17)(10,13)(11,14), (2,8)(3,9)(5,17)(6,18)(11,14)(12,15), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,7)(2,8)(3,9)(4,16)(5,17)(6,18), (4,16)(5,17)(6,18)(10,13)(11,14)(12,15), (1,16,10)(2,17,11)(3,18,12)(4,13,7)(5,14,8)(6,15,9) );

G=PermutationGroup([[(1,7),(2,8),(3,9),(4,16),(5,17),(6,18),(10,13),(11,14),(12,15)], [(1,7),(2,8),(4,16),(5,17),(10,13),(11,14)], [(2,8),(3,9),(5,17),(6,18),(11,14),(12,15)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,7),(2,8),(3,9),(4,16),(5,17),(6,18)], [(4,16),(5,17),(6,18),(10,13),(11,14),(12,15)], [(1,16,10),(2,17,11),(3,18,12),(4,13,7),(5,14,8),(6,15,9)]])

G:=TransitiveGroup(18,109);

►On 24 points - transitive group 24T579

Generators in S₂₄

(1 2)(3 5)(4 6)(7 13)(8 14)(9 15)(10 17)(11 18)(12 16)(19 22)(20 23)(21 24)

(1 7)(2 13)(3 24)(4 11)(5 21)(6 18)(8 9)(10 12)(14 15)(16 17)(19 20)(22 23)

(1 8)(2 14)(3 22)(4 12)(5 19)(6 16)(7 9)(10 11)(13 15)(17 18)(20 21)(23 24)

(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)

(1 2)(3 5)(7 13)(8 14)(9 15)(19 22)(20 23)(21 24)

(3 5)(4 6)(10 17)(11 18)(12 16)(19 22)(20 23)(21 24)

(1 5 4)(2 3 6)(7 21 11)(8 19 12)(9 20 10)(13 24 18)(14 22 16)(15 23 17)

G:=sub<Sym(24)| (1,2)(3,5)(4,6)(7,13)(8,14)(9,15)(10,17)(11,18)(12,16)(19,22)(20,23)(21,24), (1,7)(2,13)(3,24)(4,11)(5,21)(6,18)(8,9)(10,12)(14,15)(16,17)(19,20)(22,23), (1,8)(2,14)(3,22)(4,12)(5,19)(6,16)(7,9)(10,11)(13,15)(17,18)(20,21)(23,24), (7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,2)(3,5)(7,13)(8,14)(9,15)(19,22)(20,23)(21,24), (3,5)(4,6)(10,17)(11,18)(12,16)(19,22)(20,23)(21,24), (1,5,4)(2,3,6)(7,21,11)(8,19,12)(9,20,10)(13,24,18)(14,22,16)(15,23,17)>;

G:=Group( (1,2)(3,5)(4,6)(7,13)(8,14)(9,15)(10,17)(11,18)(12,16)(19,22)(20,23)(21,24), (1,7)(2,13)(3,24)(4,11)(5,21)(6,18)(8,9)(10,12)(14,15)(16,17)(19,20)(22,23), (1,8)(2,14)(3,22)(4,12)(5,19)(6,16)(7,9)(10,11)(13,15)(17,18)(20,21)(23,24), (7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,2)(3,5)(7,13)(8,14)(9,15)(19,22)(20,23)(21,24), (3,5)(4,6)(10,17)(11,18)(12,16)(19,22)(20,23)(21,24), (1,5,4)(2,3,6)(7,21,11)(8,19,12)(9,20,10)(13,24,18)(14,22,16)(15,23,17) );

G=PermutationGroup([[(1,2),(3,5),(4,6),(7,13),(8,14),(9,15),(10,17),(11,18),(12,16),(19,22),(20,23),(21,24)], [(1,7),(2,13),(3,24),(4,11),(5,21),(6,18),(8,9),(10,12),(14,15),(16,17),(19,20),(22,23)], [(1,8),(2,14),(3,22),(4,12),(5,19),(6,16),(7,9),(10,11),(13,15),(17,18),(20,21),(23,24)], [(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,2),(3,5),(7,13),(8,14),(9,15),(19,22),(20,23),(21,24)], [(3,5),(4,6),(10,17),(11,18),(12,16),(19,22),(20,23),(21,24)], [(1,5,4),(2,3,6),(7,21,11),(8,19,12),(9,20,10),(13,24,18),(14,22,16),(15,23,17)]])

G:=TransitiveGroup(24,579);

►On 24 points - transitive group 24T684

Generators in S₂₄

(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)

(1 15)(2 16)(3 12)(4 20)(5 9)(6 24)(7 19)(8 22)(10 18)(11 13)(14 17)(21 23)

(1 10)(2 13)(3 17)(4 22)(5 21)(6 7)(8 20)(9 23)(11 16)(12 14)(15 18)(19 24)

(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)

(1 10)(2 11)(3 12)(4 22)(5 23)(6 24)(7 19)(8 20)(9 21)(13 16)(14 17)(15 18)

(1 15)(2 13)(3 14)(4 20)(5 21)(6 19)(7 24)(8 22)(9 23)(10 18)(11 16)(12 17)

(1 2 3)(4 5 6)(7 22 21)(8 23 19)(9 24 20)(10 13 17)(11 14 18)(12 15 16)

G:=sub<Sym(24)| (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,15)(2,16)(3,12)(4,20)(5,9)(6,24)(7,19)(8,22)(10,18)(11,13)(14,17)(21,23), (1,10)(2,13)(3,17)(4,22)(5,21)(6,7)(8,20)(9,23)(11,16)(12,14)(15,18)(19,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,10)(2,11)(3,12)(4,22)(5,23)(6,24)(7,19)(8,20)(9,21)(13,16)(14,17)(15,18), (1,15)(2,13)(3,14)(4,20)(5,21)(6,19)(7,24)(8,22)(9,23)(10,18)(11,16)(12,17), (1,2,3)(4,5,6)(7,22,21)(8,23,19)(9,24,20)(10,13,17)(11,14,18)(12,15,16)>;

G:=Group( (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,15)(2,16)(3,12)(4,20)(5,9)(6,24)(7,19)(8,22)(10,18)(11,13)(14,17)(21,23), (1,10)(2,13)(3,17)(4,22)(5,21)(6,7)(8,20)(9,23)(11,16)(12,14)(15,18)(19,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,10)(2,11)(3,12)(4,22)(5,23)(6,24)(7,19)(8,20)(9,21)(13,16)(14,17)(15,18), (1,15)(2,13)(3,14)(4,20)(5,21)(6,19)(7,24)(8,22)(9,23)(10,18)(11,16)(12,17), (1,2,3)(4,5,6)(7,22,21)(8,23,19)(9,24,20)(10,13,17)(11,14,18)(12,15,16) );

G=PermutationGroup([[(1,4),(2,5),(3,6),(7,17),(8,18),(9,16),(10,22),(11,23),(12,24),(13,21),(14,19),(15,20)], [(1,15),(2,16),(3,12),(4,20),(5,9),(6,24),(7,19),(8,22),(10,18),(11,13),(14,17),(21,23)], [(1,10),(2,13),(3,17),(4,22),(5,21),(6,7),(8,20),(9,23),(11,16),(12,14),(15,18),(19,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,10),(2,11),(3,12),(4,22),(5,23),(6,24),(7,19),(8,20),(9,21),(13,16),(14,17),(15,18)], [(1,15),(2,13),(3,14),(4,20),(5,21),(6,19),(7,24),(8,22),(9,23),(10,18),(11,16),(12,17)], [(1,2,3),(4,5,6),(7,22,21),(8,23,19),(9,24,20),(10,13,17),(11,14,18),(12,15,16)]])

G:=TransitiveGroup(24,684);

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	3F	3G	3H	6A	6B	6C	6D	6E	···	6L	6M	6N	6O	6P
order	1	2	2	2	2	2	2	2	3	3	3	3	3	3	3	3	6	6	6	6	6	···	6	6	6	6	6
size	1	1	3	3	3	3	9	9	4	4	4	4	16	16	16	16	4	4	4	4	12	···	12	16	16	16	16

32 irreducible representations

dim	1	1	1	1	1	1	3	3	3	3	9	9
type	+	+					+	+			+	+
image	C₁	C₂	C₃	C₃	C₆	C₆	A₄	C₂×A₄	C₃×A₄	C₆×A₄	A₄²	C₂×A₄²
kernel	C₂×A₄²	A₄²	C₂³×A₄	C₂×C₂²⋊A₄	C₂²×A₄	C₂²⋊A₄	C₂×A₄	A₄	C₂³	C₂²	C₂	C₁
# reps	1	1	4	4	4	4	2	2	4	4	1	1

Matrix representation of C₂×A₄² ►in GL₇(𝔽₇)

6	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	0	1	6
0	0	0	0	1	0	6
0	0	0	0	0	0	6

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	0	6	1
0	0	0	0	0	6	0
0	0	0	0	1	6	0

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

1	0	0	0	0	0	0
0	0	0	1	0	0	0
0	6	6	6	0	0	0
0	1	0	0	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

1	0	0	0	0	0	0
0	0	1	0	0	0	0
0	1	0	0	0	0	0
0	6	6	6	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

2	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	0	1	0	0	0
0	6	6	6	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

G:=sub<GL(7,GF(7))| [6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,6,6,6],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,6,6,6,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,6,1,0,0,0,0,0,6,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,6,0,0,0,0,1,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,0,1,0,6,0,0,0,0,0,0,6,0,0,0,0,0,1,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;

C₂×A₄² in GAP, Magma, Sage, TeX

C_2\times A_4^2

% in TeX

G:=Group("C2xA4^2");

// GroupNames label

G:=SmallGroup(288,1029);

// by ID

G=gap.SmallGroup(288,1029);

# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,269,123,4548,1777]);

// Polycyclic

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

// generators/relations

G = C2×A42 order 288 = 25·32

Direct product of C2, A4 and A4

G = C₂×A₄² order 288 = 2⁵·3²

Direct product of C₂, A₄ and A₄