direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2xC3:D4, C6:2D4, C23:2S3, C22:3D6, D6:3C22, C6.10C23, Dic3:2C22, C3:3(C2xD4), (C2xC6):3C22, (C22xC6):2C2, (C22xS3):3C2, (C2xDic3):4C2, C2.10(C22xS3), SmallGroup(48,43)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xC3:D4
G = < a,b,c,d | a2=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 108 in 54 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2xC4, D4, C23, C23, Dic3, D6, D6, C2xC6, C2xC6, C2xC6, C2xD4, C2xDic3, C3:D4, C22xS3, C22xC6, C2xC3:D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:D4, C22xS3, C2xC3:D4
Character table of C2xC3:D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 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 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -√-3 | -√-3 | √-3 | 1 | 1 | -1 | √-3 | complex lifted from C3:D4 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -√-3 | √-3 | √-3 | -1 | 1 | 1 | -√-3 | complex lifted from C3:D4 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | √-3 | -√-3 | -√-3 | -1 | 1 | 1 | √-3 | complex lifted from C3:D4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | √-3 | √-3 | -√-3 | 1 | 1 | -1 | -√-3 | complex lifted from C3:D4 |
►On 24 points - transitive group 24T25
(1 5)(2 6)(3 7)(4 8)(9 23)(10 24)(11 21)(12 22)(13 18)(14 19)(15 20)(16 17)
(1 20 21)(2 22 17)(3 18 23)(4 24 19)(5 15 11)(6 12 16)(7 13 9)(8 10 14)
(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 7)(2 6)(3 5)(4 8)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)
G:=sub<Sym(24)| (1,5)(2,6)(3,7)(4,8)(9,23)(10,24)(11,21)(12,22)(13,18)(14,19)(15,20)(16,17), (1,20,21)(2,22,17)(3,18,23)(4,24,19)(5,15,11)(6,12,16)(7,13,9)(8,10,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,7)(2,6)(3,5)(4,8)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22)>;
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,23)(10,24)(11,21)(12,22)(13,18)(14,19)(15,20)(16,17), (1,20,21)(2,22,17)(3,18,23)(4,24,19)(5,15,11)(6,12,16)(7,13,9)(8,10,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,7)(2,6)(3,5)(4,8)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22) );
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,23),(10,24),(11,21),(12,22),(13,18),(14,19),(15,20),(16,17)], [(1,20,21),(2,22,17),(3,18,23),(4,24,19),(5,15,11),(6,12,16),(7,13,9),(8,10,14)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,7),(2,6),(3,5),(4,8),(9,20),(10,19),(11,18),(12,17),(13,21),(14,24),(15,23),(16,22)]])
G:=TransitiveGroup(24,25);
►On 24 points - transitive group 24T45
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 19)(14 20)(15 17)(16 18)
(1 6 19)(2 20 7)(3 8 17)(4 18 5)(9 23 13)(10 14 24)(11 21 15)(12 16 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 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)
G:=sub<Sym(24)| (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,19)(14,20)(15,17)(16,18), (1,6,19)(2,20,7)(3,8,17)(4,18,5)(9,23,13)(10,14,24)(11,21,15)(12,16,22), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,19)(14,20)(15,17)(16,18), (1,6,19)(2,20,7)(3,8,17)(4,18,5)(9,23,13)(10,14,24)(11,21,15)(12,16,22), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23) );
G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,22),(6,23),(7,24),(8,21),(13,19),(14,20),(15,17),(16,18)], [(1,6,19),(2,20,7),(3,8,17),(4,18,5),(9,23,13),(10,14,24),(11,21,15),(12,16,22)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11),(13,16),(14,15),(17,20),(18,19),(21,24),(22,23)]])
G:=TransitiveGroup(24,45);
C2xC3:D4 is a maximal subgroup of
C23.6D6 Dic3:4D4 D6:D4 C23.9D6 Dic3:D4 C23.11D6 C23.21D6 C23.28D6 C12:7D4 C23:2D6 D6:3D4 C23.14D6 C12:3D4 C24:4S3 C2xS3xD4 D4:6D6 C23.16S4
C2xC3:D4 is a maximal quotient of
C12.48D4 C23.28D6 C12:7D4 D12:6C22 C23.23D6 C23.12D6 C23:2D6 D6:3D4 C23.14D6 C12:3D4 Q8.11D6 Dic3:Q8 D6:3Q8 C12.23D4 D4:D6 Q8.13D6 Q8.14D6 C24:4S3
Matrix representation of C2xC3:D4 ►in GL3(F13) generated by
| 12 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 12 | 12 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 11 | 9 |
| 0 | 11 | 2 |
| 12 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 12 | 12 |
G:=sub<GL(3,GF(13))| [12,0,0,0,1,0,0,0,1],[1,0,0,0,12,1,0,12,0],[1,0,0,0,11,11,0,9,2],[12,0,0,0,1,12,0,0,12] >;
C2xC3:D4 in GAP, Magma, Sage, TeX
C_2\times C_3\rtimes D_4
% in TeX
G:=Group("C2xC3:D4"); // GroupNames label
G:=SmallGroup(48,43);
// by ID
G=gap.SmallGroup(48,43);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,182,804]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
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