metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.8D4, C23.2Dic3, (C2×C4).3D6, (C6×D4).2C2, (C2×D4).2S3, C3⋊2(C4.D4), C4.Dic3⋊3C2, (C22×C6).2C4, C4.13(C3⋊D4), C6.14(C22⋊C4), (C2×C12).17C22, C22.2(C2×Dic3), C2.4(C6.D4), (C2×C6).28(C2×C4), SmallGroup(96,40)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.D4
G = < a,b,c | a12=1, b4=a6, c2=a9, bab-1=a-1, cac-1=a5, cbc-1=a3b3 >
Character table of C12.D4
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | |
| ρ1 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | i | -i | -i | i | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -i | i | i | -i | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -i | -i | i | i | -1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | i | i | -i | -i | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | -2 | 2 | -1 | -2 | -2 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ14 | 2 | 2 | 2 | 2 | -2 | -1 | -2 | -2 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | -2 | 1 | 1 | -1 | -√-3 | √-3 | √-3 | -√-3 | 0 | 0 | 0 | 0 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | -2 | 1 | 1 | -1 | √-3 | -√-3 | -√-3 | √-3 | 0 | 0 | 0 | 0 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | 1 | 1 | -1 | √-3 | √-3 | -√-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | 1 | 1 | -1 | -√-3 | -√-3 | √-3 | √-3 | 0 | 0 | 0 | 0 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.D4 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | -2√-3 | 2√-3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 2√-3 | -2√-3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T99
(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 13 10 16 7 19 4 22)(2 24 11 15 8 18 5 21)(3 23 12 14 9 17 6 20)
(1 16 10 13 7 22 4 19)(2 21 11 18 8 15 5 24)(3 14 12 23 9 20 6 17)
G:=sub<Sym(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), (1,13,10,16,7,19,4,22)(2,24,11,15,8,18,5,21)(3,23,12,14,9,17,6,20), (1,16,10,13,7,22,4,19)(2,21,11,18,8,15,5,24)(3,14,12,23,9,20,6,17)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,13,10,16,7,19,4,22)(2,24,11,15,8,18,5,21)(3,23,12,14,9,17,6,20), (1,16,10,13,7,22,4,19)(2,21,11,18,8,15,5,24)(3,14,12,23,9,20,6,17) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,13,10,16,7,19,4,22),(2,24,11,15,8,18,5,21),(3,23,12,14,9,17,6,20)], [(1,16,10,13,7,22,4,19),(2,21,11,18,8,15,5,24),(3,14,12,23,9,20,6,17)]])
G:=TransitiveGroup(24,99);
C12.D4 is a maximal subgroup of
C3⋊C2≀C4 (C2×D4).D6 C24⋊5Dic3 (C22×C12)⋊C4 S3×C4.D4 M4(2).19D6 C42⋊7D6 C42⋊8D6 C24.23D4 C24.44D4 M4(2).D6 M4(2).13D6 (C6×D4).16C4 2+ 1+4⋊6S3 2+ 1+4.4S3 C36.D4 C12.D12 (C6×D4).S3 C60.28D4 C60.8D4 C5⋊(C12.D4)
C12.D4 is a maximal quotient of
C24.3Dic3 C12.(C4⋊C4) C42.7D6 C12.9D8 C12.5Q16 C36.D4 C12.D12 (C6×D4).S3 C60.28D4 C60.8D4 C5⋊(C12.D4)
Matrix representation of C12.D4 ►in GL4(𝔽7) generated by
| 1 | 0 | 6 | 2 |
| 2 | 1 | 0 | 1 |
| 5 | 5 | 5 | 4 |
| 4 | 3 | 5 | 0 |
| 6 | 0 | 3 | 2 |
| 0 | 5 | 1 | 2 |
| 4 | 3 | 5 | 2 |
| 1 | 1 | 3 | 5 |
| 2 | 3 | 0 | 1 |
| 4 | 2 | 4 | 1 |
| 4 | 3 | 5 | 5 |
| 1 | 1 | 3 | 5 |
G:=sub<GL(4,GF(7))| [1,2,5,4,0,1,5,3,6,0,5,5,2,1,4,0],[6,0,4,1,0,5,3,1,3,1,5,3,2,2,2,5],[2,4,4,1,3,2,3,1,0,4,5,3,1,1,5,5] >;
C12.D4 in GAP, Magma, Sage, TeX
C_{12}.D_4 % in TeX
G:=Group("C12.D4"); // GroupNames label
G:=SmallGroup(96,40);
// by ID
G=gap.SmallGroup(96,40);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,188,86,579,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^4=a^6,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b^3>;
// generators/relations
Export
Subgroup lattice of C12.D4 in TeX
Character table of C12.D4 in TeX