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## G = C2×A42order 288 = 25·32

### Direct product of C2, A4 and A4

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

 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 [×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 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

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