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G = (C6×C12)⋊C4  order 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⋊C4⋊2C2, C32⋊2(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

Derived series
C1 — C62 — (C6×C12)⋊C4
Chief series
C1 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — C62⋊C4 — (C6×C12)⋊C4
Lower central
C32 — C3×C6 — C62 — (C6×C12)⋊C4
Upper central
C1 — C2 — C22 — C2×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, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, C23, C32, C12, D6, C2×C6, C22⋊C4, C2×D4, C3⋊S3, C3×C6, C3×C6, D12, C2×C12, C22×S3, C23⋊C4, C3×C12, C32⋊C4, C2×C3⋊S3, C2×C3⋊S3, C62, C2×D12, C12⋊S3, C6×C12, C2×C32⋊C4, C22×C3⋊S3, C62⋊C4, C2×C12⋊S3, (C6×C12)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, 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-√32√3-√300√3√3-2√3    orthogonal faithful
ρ214-400001-20000000-32-130√32√3√3-√30-2√3-√3    orthogonal faithful
ρ224-40000-2100000-330-120-√30√3-2√32√3√3-√30    orthogonal faithful
ρ234-400001-20000000-32-130-√3-2√3-√3√302√3√3    orthogonal faithful
ρ244-40000-2100000-330-120√30-√32√3-2√3-√3√30    orthogonal faithful
ρ254-400001-2000000032-1-3-2√3-√30√3-√32√30√3    orthogonal faithful
ρ264-400001-2000000032-1-32√3√30-√3√3-2√30-√3    orthogonal faithful
ρ274-40000-21000003-30-120√3-2√3√300-√3-√32√3    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 22)(2 17 12 15)(3 24 11 20)(4 19 10 13)(5 14 9 18)(6 21 8 23)(7 16)
 
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), (1,22)(2,17,12,15)(3,24,11,20)(4,19,10,13)(5,14,9,18)(6,21,8,23)(7,16)>;
 
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), (1,22)(2,17,12,15)(3,24,11,20)(4,19,10,13)(5,14,9,18)(6,21,8,23)(7,16) );
 
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)], [(1,22),(2,17,12,15),(3,24,11,20),(4,19,10,13),(5,14,9,18),(6,21,8,23),(7,16)]])
 
G:=TransitiveGroup(24,622);
 

►On 24 points - transitive group 24T623
Generators in S24
(1 10 7 3 12 5)(2 11 8 4 9 6)(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 18 4 21)(2 15 3 24)(5 16 9 23)(6 13 12 14)(7 22 11 17)(8 19 10 20)
 
G:=sub<Sym(24)| (1,10,7,3,12,5)(2,11,8,4,9,6)(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), (1,18,4,21)(2,15,3,24)(5,16,9,23)(6,13,12,14)(7,22,11,17)(8,19,10,20)>;
 
G:=Group( (1,10,7,3,12,5)(2,11,8,4,9,6)(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), (1,18,4,21)(2,15,3,24)(5,16,9,23)(6,13,12,14)(7,22,11,17)(8,19,10,20) );
 
G=PermutationGroup([[(1,10,7,3,12,5),(2,11,8,4,9,6),(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)], [(1,18,4,21),(2,15,3,24),(5,16,9,23),(6,13,12,14),(7,22,11,17),(8,19,10,20)]])
 
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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