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G = (C6×C12)⋊5C4order 288 = 25·32

5th semidirect product of C6×C12 and C4 acting faithfully

metabelian, soluble, monomial

Aliases: (C6×C12)⋊5C4, C62.18(C2×C4), C62⋊C4.2C2, C321(C42⋊C2), (C4×C3⋊S3)⋊10C4, (C4×C32⋊C4)⋊8C2, C4⋊(C32⋊C4)⋊6C2, (C2×C4)⋊4(C32⋊C4), C4.14(C2×C32⋊C4), (C3×C12).21(C2×C4), (C2×C3⋊Dic3)⋊14C4, C22.7(C2×C32⋊C4), C3⋊S3.10(C4○D4), C2.7(C22×C32⋊C4), (C2×C3⋊S3).36C23, C3⋊Dic3.53(C2×C4), (C3×C6).29(C22×C4), (C4×C3⋊S3).100C22, (C2×C32⋊C4).23C22, (C22×C3⋊S3).97C22, (C2×C4×C3⋊S3).30C2, (C2×C3⋊S3).49(C2×C4), SmallGroup(288,934)

Series: Derived Chief Lower central Upper central

C1C3×C6 — (C6×C12)⋊5C4
C1C32C3⋊S3C2×C3⋊S3C2×C32⋊C4C4×C32⋊C4 — (C6×C12)⋊5C4
C32C3×C6 — (C6×C12)⋊5C4
C1C4C2×C4

Generators and relations for (C6×C12)⋊5C4
 G = < a,b,c | a6=b12=c4=1, ab=ba, cac-1=a-1b2, cbc-1=a2b >

Subgroups: 656 in 130 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2 [×4], C3 [×2], C4 [×2], C4 [×6], C22, C22 [×4], S3 [×8], C6 [×6], C2×C4, C2×C4 [×9], C23, C32, Dic3 [×4], C12 [×4], D6 [×12], C2×C6 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6, C4×S3 [×8], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×2], C42⋊C2, C3⋊Dic3 [×2], C3×C12 [×2], C32⋊C4 [×4], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, S3×C2×C4 [×2], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C2×C32⋊C4 [×4], C22×C3⋊S3, C4×C32⋊C4 [×2], C4⋊(C32⋊C4) [×2], C62⋊C4 [×2], C2×C4×C3⋊S3, (C6×C12)⋊5C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C4○D4 [×2], C42⋊C2, C32⋊C4, C2×C32⋊C4 [×3], C22×C32⋊C4, (C6×C12)⋊5C4

Permutation representations of (C6×C12)⋊5C4
On 24 points - transitive group 24T621
Generators in S24
(13 15 17 19 21 23)(14 16 18 20 22 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 16)(2 17 6 21)(3 18 11 14)(4 19)(5 20 9 24)(7 22)(8 23 12 15)(10 13)

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

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

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

G:=TransitiveGroup(24,621);

36 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F···4N6A···6F12A···12H
order12222233444444···46···612···12
size1129918441129918···184···44···4

36 irreducible representations

dim1111111124444
type++++++++
imageC1C2C2C2C2C4C4C4C4○D4C32⋊C4C2×C32⋊C4C2×C32⋊C4(C6×C12)⋊5C4
kernel(C6×C12)⋊5C4C4×C32⋊C4C4⋊(C32⋊C4)C62⋊C4C2×C4×C3⋊S3C4×C3⋊S3C2×C3⋊Dic3C6×C12C3⋊S3C2×C4C4C22C1
# reps1222142242428

Matrix representation of (C6×C12)⋊5C4 in GL4(𝔽5) generated by

0112
1322
0204
0242
,
3333
4120
2230
1330
,
0112
0243
3112
1242
G:=sub<GL(4,GF(5))| [0,1,0,0,1,3,2,2,1,2,0,4,2,2,4,2],[3,4,2,1,3,1,2,3,3,2,3,3,3,0,0,0],[0,0,3,1,1,2,1,2,1,4,1,4,2,3,2,2] >;

(C6×C12)⋊5C4 in GAP, Magma, Sage, TeX

(C_6\times C_{12})\rtimes_5C_4
% in TeX

G:=Group("(C6xC12):5C4");
// GroupNames label

G:=SmallGroup(288,934);
// by ID

G=gap.SmallGroup(288,934);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,120,422,9413,362,12550,1203]);
// Polycyclic

G:=Group<a,b,c|a^6=b^12=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b^2,c*b*c^-1=a^2*b>;
// generators/relations

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