non-abelian, soluble, monomial, rational
Aliases: C4⋊1S3≀C2, (C3×C12)⋊2D4, C32⋊C4⋊1D4, C3⋊Dic3⋊6D4, C32⋊(C4⋊1D4), D6⋊D6⋊9C2, (C2×S3≀C2)⋊2C2, (C4×C32⋊C4)⋊4C2, C3⋊S3.3(C2×D4), C2.14(C2×S3≀C2), (C2×S32).2C22, (C3×C6).12(C2×D4), (C2×C3⋊S3).6C23, (C4×C3⋊S3).36C22, (C2×C32⋊C4).15C22, SmallGroup(288,879)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C32 — C2×C3⋊S3 — (C3×C12)⋊D4 |
C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×S32 — C2×S3≀C2 — (C3×C12)⋊D4 |
C32 — C2×C3⋊S3 — (C3×C12)⋊D4 |
Generators and relations for (C3×C12)⋊D4
G = < a,b,c,d | a3=b12=c4=d2=1, ab=ba, cac-1=b4, dad=a-1, cbc-1=a-1b9, dbd=b7, dcd=c-1 >
Subgroups: 1048 in 162 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2 [×6], C3 [×2], C4, C4 [×5], C22 [×13], S3 [×8], C6 [×6], C2×C4 [×3], D4 [×12], C23 [×4], C32, Dic3 [×2], C12 [×2], D6 [×14], C2×C6 [×4], C42, C2×D4 [×6], C3×S3 [×4], C3⋊S3 [×2], C3×C6, C4×S3 [×2], D12 [×2], C3⋊D4 [×4], C3×D4 [×2], C22×S3 [×4], C4⋊1D4, C3⋊Dic3, C3×C12, C32⋊C4 [×4], S32 [×8], S3×C6 [×4], C2×C3⋊S3, S3×D4 [×2], D6⋊S3 [×2], C3×D12 [×2], C4×C3⋊S3, S3≀C2 [×8], C2×C32⋊C4 [×2], C2×S32 [×4], C4×C32⋊C4, D6⋊D6 [×2], C2×S3≀C2 [×4], (C3×C12)⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C4⋊1D4, S3≀C2, C2×S3≀C2, (C3×C12)⋊D4
Character table of (C3×C12)⋊D4
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | |
size | 1 | 1 | 9 | 9 | 12 | 12 | 12 | 12 | 4 | 4 | 2 | 18 | 18 | 18 | 18 | 18 | 4 | 4 | 24 | 24 | 24 | 24 | 8 | 8 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
ρ5 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | linear of order 2 |
ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
ρ9 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | orthogonal lifted from D4 |
ρ10 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ11 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
ρ13 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ15 | 4 | 4 | 0 | 0 | 0 | 0 | 2 | -2 | 1 | -2 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | -1 | 0 | 1 | 2 | -1 | orthogonal lifted from C2×S3≀C2 |
ρ16 | 4 | 4 | 0 | 0 | 0 | 0 | 2 | 2 | 1 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | -1 | 0 | -1 | -2 | 1 | orthogonal lifted from S3≀C2 |
ρ17 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | 2 | 1 | -2 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 1 | 0 | -1 | 2 | -1 | orthogonal lifted from C2×S3≀C2 |
ρ18 | 4 | 4 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | 0 | -1 | 0 | -1 | 2 | orthogonal lifted from C2×S3≀C2 |
ρ19 | 4 | 4 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | 1 | 4 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | -1 | 0 | -1 | 0 | 1 | -2 | orthogonal lifted from S3≀C2 |
ρ20 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 1 | 0 | 1 | -2 | 1 | orthogonal lifted from S3≀C2 |
ρ21 | 4 | 4 | 0 | 0 | -2 | -2 | 0 | 0 | -2 | 1 | 4 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | 0 | 1 | 0 | 1 | -2 | orthogonal lifted from S3≀C2 |
ρ22 | 4 | 4 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | -1 | 0 | 1 | 0 | -1 | 2 | orthogonal lifted from C2×S3≀C2 |
ρ23 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
ρ24 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 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 22)(2 15 6 23)(3 20 11 24)(4 13)(5 18 9 14)(7 16)(8 21 12 17)(10 19)
(1 22)(2 17)(3 24)(4 19)(5 14)(6 21)(7 16)(8 23)(9 18)(10 13)(11 20)(12 15)
G:=sub<Sym(24)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,22)(2,15,6,23)(3,20,11,24)(4,13)(5,18,9,14)(7,16)(8,21,12,17)(10,19), (1,22)(2,17)(3,24)(4,19)(5,14)(6,21)(7,16)(8,23)(9,18)(10,13)(11,20)(12,15)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,22)(2,15,6,23)(3,20,11,24)(4,13)(5,18,9,14)(7,16)(8,21,12,17)(10,19), (1,22)(2,17)(3,24)(4,19)(5,14)(6,21)(7,16)(8,23)(9,18)(10,13)(11,20)(12,15) );
G=PermutationGroup([(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,22),(2,15,6,23),(3,20,11,24),(4,13),(5,18,9,14),(7,16),(8,21,12,17),(10,19)], [(1,22),(2,17),(3,24),(4,19),(5,14),(6,21),(7,16),(8,23),(9,18),(10,13),(11,20),(12,15)])
G:=TransitiveGroup(24,653);
Matrix representation of (C3×C12)⋊D4 ►in GL6(ℤ)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | -1 | 1 | 0 | 0 |
0 | 0 | -1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | -1 | -1 |
0 | 0 | -1 | 0 | 1 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
-1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | -1 | 1 | 0 | 0 |
0 | 0 | -1 | 0 | 0 | 0 |
0 | 0 | -1 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | -1 | -1 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | -1 | 1 |
0 | 0 | 1 | 1 | -2 | -1 |
0 | 0 | 0 | 0 | -1 | 0 |
0 | 0 | 0 | 1 | -1 | 0 |
0 | -1 | 0 | 0 | 0 | 0 |
-1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | -1 | 1 |
0 | 0 | 1 | 1 | -2 | -1 |
0 | 0 | 0 | 0 | -1 | 0 |
0 | 0 | 1 | 0 | -1 | 0 |
G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,-1,0,0,1,0,1,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,-1,0,0,0,1,0,0,1,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,-1,-2,-1,-1,0,0,1,-1,0,0],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,-1,-2,-1,-1,0,0,1,-1,0,0] >;
(C3×C12)⋊D4 in GAP, Magma, Sage, TeX
(C_3\times C_{12})\rtimes D_4
% in TeX
G:=Group("(C3xC12):D4");
// GroupNames label
G:=SmallGroup(288,879);
// by ID
G=gap.SmallGroup(288,879);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,422,219,100,2693,2028,362,797,1203]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=c^4=d^2=1,a*b=b*a,c*a*c^-1=b^4,d*a*d=a^-1,c*b*c^-1=a^-1*b^9,d*b*d=b^7,d*c*d=c^-1>;
// generators/relations