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G = C2×A42order 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⋊A42C6, C221(C6×A4), C233(C3×A4), (C22×A4)⋊2C6, (C2×C22⋊A4)⋊C3, SmallGroup(288,1029)

Series: Derived Chief Lower central Upper central

C1C24 — C2×A42
C1C22C24C22×A4A42 — C2×A42
C24 — C2×A42
C1C2

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 [×6], C3 [×4], C22 [×2], C22 [×21], C6 [×8], C23 [×2], C23 [×21], C32, A4 [×2], A4 [×6], C2×C6 [×6], C24, C24 [×6], C3×C6, C2×A4 [×2], C2×A4 [×10], C22×C6 [×2], C25, C3×A4 [×2], C22×A4 [×2], C22×A4 [×4], C22⋊A4 [×2], C6×A4 [×2], C23×A4 [×2], C2×C22⋊A4 [×2], A42, C2×A42
Quotients: C1, C2, C3 [×4], C6 [×4], C32, A4 [×2], C3×C6, C2×A4 [×2], C3×A4 [×2], C6×A4 [×2], 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 4)(2 6)(3 5)(7 23)(8 24)(9 22)(10 13)(11 14)(12 15)(16 20)(17 21)(18 19)
(1 20)(2 8)(3 12)(4 16)(5 15)(6 24)(7 9)(10 11)(13 14)(17 18)(19 21)(22 23)
(1 21)(2 9)(3 10)(4 17)(5 13)(6 22)(7 8)(11 12)(14 15)(16 18)(19 20)(23 24)
(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(2 6)(3 5)(7 23)(8 24)(9 22)(10 13)(11 14)(12 15)
(1 4)(3 5)(10 13)(11 14)(12 15)(16 20)(17 21)(18 19)
(1 6 5)(2 3 4)(7 11 18)(8 12 16)(9 10 17)(13 21 22)(14 19 23)(15 20 24)

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

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

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

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 2A2B2C2D2E2F2G3A3B3C3D3E3F3G3H6A6B6C6D6E···6L6M6N6O6P
order122222223333333366666···66666
size11333399444416161616444412···1216161616

32 irreducible representations

dim111111333399
type++++++
imageC1C2C3C3C6C6A4C2×A4C3×A4C6×A4A42C2×A42
kernelC2×A42A42C23×A4C2×C22⋊A4C22×A4C22⋊A4C2×A4A4C23C22C2C1
# reps114444224411

Matrix representation of C2×A42 in GL7(𝔽7)

6000000
0100000
0010000
0001000
0000100
0000010
0000001
,
1000000
0100000
0010000
0001000
0000016
0000106
0000006
,
1000000
0100000
0010000
0001000
0000061
0000060
0000160
,
1000000
0100000
0010000
0001000
0000010
0000001
0000100
,
1000000
0001000
0666000
0100000
0000100
0000010
0000001
,
1000000
0010000
0100000
0666000
0000100
0000010
0000001
,
2000000
0100000
0001000
0666000
0000100
0000010
0000001

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

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