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## G = C2×C62⋊C4order 288 = 25·32

### Direct product of C2 and C62⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C62⋊C4
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×C32⋊C4 — C22×C32⋊C4 — C2×C62⋊C4
 Lower central C32 — C3×C6 — C2×C62⋊C4
 Upper central C1 — C22 — C23

Generators and relations for C2×C62⋊C4
G = < a,b,c,d | a2=b6=c6=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b4c >

Subgroups: 1488 in 266 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C2×C4, C23, C23, C32, D6, C2×C6, C22⋊C4, C22×C4, C24, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C22×S3, C22×C6, C2×C22⋊C4, C32⋊C4, C2×C3⋊S3, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, S3×C23, C2×C32⋊C4, C2×C32⋊C4, C22×C3⋊S3, C22×C3⋊S3, C22×C3⋊S3, C2×C62, C62⋊C4, C22×C32⋊C4, C23×C3⋊S3, C2×C62⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C32⋊C4, C2×C32⋊C4, C62⋊C4, C22×C32⋊C4, C2×C62⋊C4

Permutation representations of C2×C62⋊C4
On 24 points - transitive group 24T674
Generators in S24
(1 6)(2 5)(3 12)(4 11)(7 10)(8 9)(13 16)(14 17)(15 18)(19 22)(20 23)(21 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 3 9 5 11 7)(2 4 10 6 12 8)(13 23 15 19 17 21)(14 24 16 20 18 22)
(1 13 7 21)(2 22 8 18)(3 23 11 17)(4 14 12 20)(5 19 9 15)(6 16 10 24)

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

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

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

G:=TransitiveGroup(24,674);

On 24 points - transitive group 24T675
Generators in S24
(1 7)(2 8)(3 9)(4 12)(5 10)(6 11)(13 19)(14 20)(15 21)(16 22)(17 23)(18 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 5)(2 6)(3 4)(7 10)(8 11)(9 12)(13 18 17 16 15 14)(19 24 23 22 21 20)
(1 13 7 19)(2 15 9 23)(3 17 8 21)(4 14 11 24)(5 16 10 22)(6 18 12 20)

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

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

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

G:=TransitiveGroup(24,675);

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3A 3B 4A ··· 4H 6A ··· 6N order 1 2 2 2 2 2 2 2 2 2 2 2 3 3 4 ··· 4 6 ··· 6 size 1 1 1 1 2 2 9 9 9 9 18 18 4 4 18 ··· 18 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 1 2 4 4 4 type + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 C32⋊C4 C2×C32⋊C4 C62⋊C4 kernel C2×C62⋊C4 C62⋊C4 C22×C32⋊C4 C23×C3⋊S3 C22×C3⋊S3 C2×C62 C2×C3⋊S3 C23 C22 C2 # reps 1 4 2 1 6 2 4 2 6 8

Matrix representation of C2×C62⋊C4 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 0 0 1 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 1
,
 0 8 0 0 0 0 5 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 1 0 0 0 0 1 0 0 0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,12,0,0] >;

C2×C62⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_6^2\rtimes C_4
% in TeX

G:=Group("C2xC6^2:C4");
// GroupNames label

G:=SmallGroup(288,941);
// by ID

G=gap.SmallGroup(288,941);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,422,9413,362,12550,1203]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^6=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^4*c>;
// generators/relations

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