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G = (C6×C12)⋊C4order 288 = 25·32

1st semidirect product of C6×C12 and C4 acting faithfully

metabelian, soluble, monomial

Aliases: (C6×C12)⋊1C4, C62.2(C2×C4), C62⋊C42C2, C322(C23⋊C4), C2.4(C62⋊C4), (C2×C4)⋊(C32⋊C4), (C22×C3⋊S3)⋊3C4, (C2×C3⋊S3).13D4, (C2×C12⋊S3).2C2, C22.2(C2×C32⋊C4), (C3×C6).12(C22⋊C4), (C22×C3⋊S3).4C22, SmallGroup(288,422)

Series: Derived Chief Lower central Upper central

C1C62 — (C6×C12)⋊C4
C1C32C3×C6C2×C3⋊S3C22×C3⋊S3C62⋊C4 — (C6×C12)⋊C4
C32C3×C6C62 — (C6×C12)⋊C4
C1C2C22C2×C4

Generators and relations for (C6×C12)⋊C4
 G = < a,b,c | a6=b12=c4=1, ab=ba, cac-1=a-1b2, cbc-1=a-1b7 >

Subgroups: 792 in 106 conjugacy classes, 16 normal (12 characteristic)
C1, C2, C2 [×4], C3 [×2], C4 [×3], C22, C22 [×5], S3 [×8], C6 [×6], C2×C4, C2×C4 [×2], D4 [×2], C23 [×2], C32, C12 [×4], D6 [×16], C2×C6 [×2], C22⋊C4 [×2], C2×D4, C3⋊S3 [×3], C3×C6, C3×C6, D12 [×8], C2×C12 [×2], C22×S3 [×4], C23⋊C4, C3×C12, C32⋊C4 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×3], C62, C2×D12 [×2], C12⋊S3 [×2], C6×C12, C2×C32⋊C4 [×2], C22×C3⋊S3 [×2], C62⋊C4 [×2], C2×C12⋊S3, (C6×C12)⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, C23⋊C4, C32⋊C4, C2×C32⋊C4, C62⋊C4, (C6×C12)⋊C4

Character table of (C6×C12)⋊C4

 class 12A2B2C2D2E3A3B4A4B4C4D4E6A6B6C6D6E6F12A12B12C12D12E12F12G12H
 size 1121818364443636363644444444444444
ρ1111111111111111111111111111    trivial
ρ211111-111-1-111-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311111-111-11-1-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ4111111111-1-1-1-111111111111111    linear of order 2
ρ5111-1-1111-1-i-iii111111-1-1-1-1-1-1-1-1    linear of order 4
ρ6111-1-1-1111i-ii-i11111111111111    linear of order 4
ρ7111-1-1-1111-ii-ii11111111111111    linear of order 4
ρ8111-1-1111-1ii-i-i111111-1-1-1-1-1-1-1-1    linear of order 4
ρ922-22-202200000-2-2-222-200000000    orthogonal lifted from D4
ρ1022-2-2202200000-2-2-222-200000000    orthogonal lifted from D4
ρ1144-4000-2100000-1-121-2230-3003-30    orthogonal lifted from C62⋊C4
ρ12444000-21-4000011-21-2-2-12-122-1-12    orthogonal lifted from C2×C32⋊C4
ρ134-400004400000000-4-4000000000    orthogonal lifted from C23⋊C4
ρ1444-40001-20000022-1-21-1030-3-3003    orthogonal lifted from C62⋊C4
ρ1544-40001-20000022-1-21-10-303300-3    orthogonal lifted from C62⋊C4
ρ1644-4000-2100000-1-121-22-30300-330    orthogonal lifted from C62⋊C4
ρ17444000-214000011-21-2-21-21-2-211-2    orthogonal lifted from C32⋊C4
ρ184440001-2-40000-2-21-2112-12-1-122-1    orthogonal lifted from C2×C32⋊C4
ρ194440001-240000-2-21-211-21-211-2-21    orthogonal lifted from C32⋊C4
ρ204-40000-21000003-30-120-323-30033-23    orthogonal faithful
ρ214-400001-20000000-32-1303233-30-23-3    orthogonal faithful
ρ224-40000-2100000-330-120-303-23233-30    orthogonal faithful
ρ234-400001-20000000-32-130-3-23-330233    orthogonal faithful
ρ244-40000-2100000-330-12030-323-23-330    orthogonal faithful
ρ254-400001-2000000032-1-3-23-303-32303    orthogonal faithful
ρ264-400001-2000000032-1-32330-33-230-3    orthogonal faithful
ρ274-40000-21000003-30-1203-23300-3-323    orthogonal faithful

Permutation representations of (C6×C12)⋊C4
On 24 points - transitive group 24T622
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 23 12 21)(3 18 11 14)(4 13 10 19)(5 20 9 24)(6 15 8 17)(7 22)

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

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

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

G:=TransitiveGroup(24,622);

On 24 points - transitive group 24T623
Generators in S24
(1 7 12 3 5 10)(2 8 9 4 6 11)(13 17 21)(14 18 22)(15 19 23)(16 20 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 20 9 21)(2 17 12 24)(3 14 11 15)(4 23 10 18)(5 16 6 13)(7 22 8 19)

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

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

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

G:=TransitiveGroup(24,623);

Matrix representation of (C6×C12)⋊C4 in GL4(𝔽13) generated by

01200
11200
118120
118012
,
101000
3700
93710
31310
,
00121
251112
97912
24912
G:=sub<GL(4,GF(13))| [0,1,11,11,12,12,8,8,0,0,12,0,0,0,0,12],[10,3,9,3,10,7,3,1,0,0,7,3,0,0,10,10],[0,2,9,2,0,5,7,4,12,11,9,9,1,12,12,12] >;

(C6×C12)⋊C4 in GAP, Magma, Sage, TeX

(C_6\times C_{12})\rtimes C_4
% in TeX

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

G:=SmallGroup(288,422);
// by ID

G=gap.SmallGroup(288,422);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,120,219,675,9413,691,12550,2372]);
// 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^-1*b^7>;
// generators/relations

Export

Character table of (C6×C12)⋊C4 in TeX

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