metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D12, C4○Dic6, D12⋊5C2, C4.16D6, Dic6⋊5C2, C6.4C23, C22.2D6, D6.1C22, C12.16C22, Dic3.2C22, (C2×C4)⋊3S3, (C4×S3)⋊4C2, (C2×C12)⋊4C2, C4○(C3⋊D4), C3⋊1(C4○D4), C3⋊D4⋊3C2, C2.5(C22×S3), (C2×C6).11C22, SmallGroup(48,37)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D12
G = < a,b,c | a4=c2=1, b6=a2, ab=ba, ac=ca, cbc=a2b5 >
Character table of C4○D12
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 2 | 6 | 6 | 2 | 1 | 1 | 2 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | -2 | 0 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | -2 | 0 | 0 | -2i | 0 | 0 | 2i | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | 0 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | -2 | 0 | 0 | 2i | 0 | 0 | -2i | complex lifted from C4○D4 |
| ρ15 | 2 | -2 | 0 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 1 | -√-3 | √-3 | -i | -√3 | √3 | i | complex faithful |
| ρ16 | 2 | -2 | 0 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 1 | √-3 | -√-3 | -i | √3 | -√3 | i | complex faithful |
| ρ17 | 2 | -2 | 0 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 1 | -√-3 | √-3 | i | √3 | -√3 | -i | complex faithful |
| ρ18 | 2 | -2 | 0 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 1 | √-3 | -√-3 | i | -√3 | √3 | -i | complex faithful |
►On 24 points - transitive group 24T19
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 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 14)(2 13)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 18)(10 17)(11 16)(12 15)
G:=sub<Sym(24)| (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,14)(2,13)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)>;
G:=Group( (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,14)(2,13)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15) );
G=PermutationGroup([[(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,16,19,22),(14,17,20,23),(15,18,21,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,14),(2,13),(3,24),(4,23),(5,22),(6,21),(7,20),(8,19),(9,18),(10,17),(11,16),(12,15)]])
G:=TransitiveGroup(24,19);
►On 24 points - transitive group 24T24
(1 23 7 17)(2 24 8 18)(3 13 9 19)(4 14 10 20)(5 15 11 21)(6 16 12 22)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)
G:=sub<Sym(24)| (1,23,7,17)(2,24,8,18)(3,13,9,19)(4,14,10,20)(5,15,11,21)(6,16,12,22), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)>;
G:=Group( (1,23,7,17)(2,24,8,18)(3,13,9,19)(4,14,10,20)(5,15,11,21)(6,16,12,22), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23) );
G=PermutationGroup([[(1,23,7,17),(2,24,8,18),(3,13,9,19),(4,14,10,20),(5,15,11,21),(6,16,12,22)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,20),(14,19),(15,18),(16,17),(21,24),(22,23)]])
G:=TransitiveGroup(24,24);
C4○D12 is a maximal subgroup of
C42⋊4S3 D12⋊C4 C8○D12 C4○D24 D12.C4 C8⋊D6 C8.D6 D12⋊6C22 Q8.11D6 Q8.13D6 D4⋊6D6 Q8.15D6 S3×C4○D4 D4○D12 Q8○D12 D36⋊5C2 D12⋊5S3 D6.D6 D6.6D6 D6.3D6 C12.59D6 C24.10D6 GL2(𝔽3)⋊C22 D6.D10 D12⋊5D5 C12.28D10 Dic3.D10 D60⋊11C2 D6.D14 D12⋊5D7 D14.D6 Dic3.D14 D84⋊11C2 C4.6S5
C4○D12 is a maximal quotient of
C4×Dic6 C12.6Q8 C42⋊2S3 C4×D12 C42⋊7S3 C42⋊3S3 C23.8D6 C23.9D6 Dic3⋊D4 C23.11D6 Dic3.Q8 D6.D4 D6⋊Q8 C4⋊C4⋊S3 C12.48D4 C23.26D6 C4×C3⋊D4 C23.28D6 C12⋊7D4 D36⋊5C2 D12⋊5S3 D6.D6 D6.6D6 D6.3D6 C12.59D6 C24.10D6 D6.D10 D12⋊5D5 C12.28D10 Dic3.D10 D60⋊11C2 D6.D14 D12⋊5D7 D14.D6 Dic3.D14 D84⋊11C2
Matrix representation of C4○D12 ►in GL2(𝔽13) generated by
| 8 | 0 |
| 0 | 8 |
| 6 | 10 |
| 3 | 3 |
| 12 | 1 |
| 0 | 1 |
G:=sub<GL(2,GF(13))| [8,0,0,8],[6,3,10,3],[12,0,1,1] >;
C4○D12 in GAP, Magma, Sage, TeX
C_4\circ D_{12} % in TeX
G:=Group("C4oD12"); // GroupNames label
G:=SmallGroup(48,37);
// by ID
G=gap.SmallGroup(48,37);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,46,182,804]);
// Polycyclic
G:=Group<a,b,c|a^4=c^2=1,b^6=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b^5>;
// generators/relations
Export
Subgroup lattice of C4○D12 in TeX
Character table of C4○D12 in TeX