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G = C2×C62⋊C4order 288 = 25·32

Direct product of C2 and C62⋊C4

direct product, metabelian, soluble, monomial

Aliases: C2×C62⋊C4, C622(C2×C4), (C2×C62)⋊5C4, C232(C32⋊C4), (C22×C3⋊S3)⋊8C4, (C2×C3⋊S3).69D4, C3⋊S3.12(C2×D4), C222(C2×C32⋊C4), C324(C2×C22⋊C4), C3⋊S34(C22⋊C4), (C3×C6)⋊2(C22⋊C4), (C23×C3⋊S3).5C2, (C22×C32⋊C4)⋊4C2, (C2×C32⋊C4)⋊3C22, (C2×C3⋊S3).39C23, (C3×C6).36(C22×C4), C2.13(C22×C32⋊C4), (C22×C3⋊S3).98C22, (C2×C3⋊S3)⋊18(C2×C4), SmallGroup(288,941)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C2×C62⋊C4
C1C32C3⋊S3C2×C3⋊S3C2×C32⋊C4C22×C32⋊C4 — C2×C62⋊C4
C32C3×C6 — C2×C62⋊C4
C1C22C23

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 [×2], C2 [×8], C3 [×2], C4 [×4], C22, C22 [×2], C22 [×20], S3 [×16], C6 [×14], C2×C4 [×8], C23, C23 [×10], C32, D6 [×56], C2×C6 [×14], C22⋊C4 [×4], C22×C4 [×2], C24, C3⋊S3 [×4], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], C22×S3 [×28], C22×C6 [×2], C2×C22⋊C4, C32⋊C4 [×4], C2×C3⋊S3 [×2], C2×C3⋊S3 [×6], C2×C3⋊S3 [×10], C62, C62 [×2], C62 [×2], S3×C23 [×2], C2×C32⋊C4 [×4], C2×C32⋊C4 [×4], C22×C3⋊S3 [×2], C22×C3⋊S3 [×4], C22×C3⋊S3 [×4], C2×C62, C62⋊C4 [×4], C22×C32⋊C4 [×2], C23×C3⋊S3, C2×C62⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C32⋊C4, C2×C32⋊C4 [×3], C62⋊C4 [×2], C22×C32⋊C4, C2×C62⋊C4

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

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

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

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

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

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

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

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

G:=TransitiveGroup(24,675);

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K3A3B4A···4H6A···6N
order122222222222334···46···6
size111122999918184418···184···4

36 irreducible representations

dim1111112444
type++++++++
imageC1C2C2C2C4C4D4C32⋊C4C2×C32⋊C4C62⋊C4
kernelC2×C62⋊C4C62⋊C4C22×C32⋊C4C23×C3⋊S3C22×C3⋊S3C2×C62C2×C3⋊S3C23C22C2
# reps1421624268

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

100000
010000
0012000
0001200
0000120
0000012
,
1200000
010000
0011200
001000
0000121
0000120
,
1200000
0120000
0012000
0001200
000001
0000121
,
080000
500000
0000120
0000012
000100
001000

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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