metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.53D4, M4(2).1S3, C22.1Dic6, (C2×C6).Q8, C3⋊C8.1C4, C6.8(C4⋊C4), C4.13(C4×S3), C12.5(C2×C4), (C2×C4).37D6, C3⋊2(C8.C4), C4.28(C3⋊D4), C4.Dic3.2C2, C2.5(Dic3⋊C4), (C2×C12).12C22, (C3×M4(2)).1C2, (C2×C3⋊C8).4C2, SmallGroup(96,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.53D4
G = < a,b,c | a12=1, b4=a6, c2=a9, bab-1=cac-1=a5, cbc-1=a6b3 >
Character table of C12.53D4
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | |
| 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 | -i | i | -1 | 1 | 1 | -1 | i | -i | -1 | -1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | 1 | -1 | -1 | 1 | i | -i | -1 | -1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | -1 | 1 | 1 | -1 | -i | i | -1 | -1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | i | 1 | -1 | -1 | 1 | -i | i | -1 | -1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | -1 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | 2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ13 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -√3 | √3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ14 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | √3 | -√3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ15 | 2 | 2 | -2 | -1 | -2 | -2 | 2 | -1 | 1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | i | i | -i | -i | complex lifted from C4×S3 |
| ρ16 | 2 | 2 | -2 | -1 | -2 | -2 | 2 | -1 | 1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -i | -i | i | i | complex lifted from C4×S3 |
| ρ17 | 2 | 2 | -2 | -1 | 2 | 2 | -2 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -√-3 | √-3 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | -1 | 2 | 2 | -2 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | √-3 | -√-3 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ19 | 2 | -2 | 0 | 2 | 2i | -2i | 0 | -2 | 0 | 0 | 0 | -√-2 | √2 | -√2 | √-2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C8.C4 |
| ρ20 | 2 | -2 | 0 | 2 | -2i | 2i | 0 | -2 | 0 | 0 | 0 | -√-2 | -√2 | √2 | √-2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C8.C4 |
| ρ21 | 2 | -2 | 0 | 2 | -2i | 2i | 0 | -2 | 0 | 0 | 0 | √-2 | √2 | -√2 | -√-2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C8.C4 |
| ρ22 | 2 | -2 | 0 | 2 | 2i | -2i | 0 | -2 | 0 | 0 | 0 | √-2 | -√2 | √2 | -√-2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C8.C4 |
| ρ23 | 4 | -4 | 0 | -2 | 4i | -4i | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | -4 | 0 | -2 | -4i | 4i | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 48 points
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 31 4 34 7 25 10 28)(2 36 5 27 8 30 11 33)(3 29 6 32 9 35 12 26)(13 43 22 40 19 37 16 46)(14 48 23 45 20 42 17 39)(15 41 24 38 21 47 18 44)
(1 43 10 40 7 37 4 46)(2 48 11 45 8 42 5 39)(3 41 12 38 9 47 6 44)(13 28 22 25 19 34 16 31)(14 33 23 30 20 27 17 36)(15 26 24 35 21 32 18 29)
G:=sub<Sym(48)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,31,4,34,7,25,10,28)(2,36,5,27,8,30,11,33)(3,29,6,32,9,35,12,26)(13,43,22,40,19,37,16,46)(14,48,23,45,20,42,17,39)(15,41,24,38,21,47,18,44), (1,43,10,40,7,37,4,46)(2,48,11,45,8,42,5,39)(3,41,12,38,9,47,6,44)(13,28,22,25,19,34,16,31)(14,33,23,30,20,27,17,36)(15,26,24,35,21,32,18,29)>;
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)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,31,4,34,7,25,10,28)(2,36,5,27,8,30,11,33)(3,29,6,32,9,35,12,26)(13,43,22,40,19,37,16,46)(14,48,23,45,20,42,17,39)(15,41,24,38,21,47,18,44), (1,43,10,40,7,37,4,46)(2,48,11,45,8,42,5,39)(3,41,12,38,9,47,6,44)(13,28,22,25,19,34,16,31)(14,33,23,30,20,27,17,36)(15,26,24,35,21,32,18,29) );
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),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,31,4,34,7,25,10,28),(2,36,5,27,8,30,11,33),(3,29,6,32,9,35,12,26),(13,43,22,40,19,37,16,46),(14,48,23,45,20,42,17,39),(15,41,24,38,21,47,18,44)], [(1,43,10,40,7,37,4,46),(2,48,11,45,8,42,5,39),(3,41,12,38,9,47,6,44),(13,28,22,25,19,34,16,31),(14,33,23,30,20,27,17,36),(15,26,24,35,21,32,18,29)]])
C12.53D4 is a maximal subgroup of
D12.2D4 D12.3D4 D12.6D4 D12.7D4 M4(2).22D6 C42.196D6 S3×C8.C4 M4(2).25D6 C23.8Dic6 C24.100D4 C24.54D4 M4(2).D6 M4(2).13D6 M4(2).15D6 M4(2).16D6 C36.53D4 C12.82D12 C62.5Q8 C62.8Q8 C60.D4 C12.59D20 C60.210D4 D10.Dic6 D10.2Dic6
C12.53D4 is a maximal quotient of
C12.53D8 C12.39SD16 C12.4C42 C36.53D4 C12.82D12 C62.5Q8 C62.8Q8 C60.D4 C12.59D20 C60.210D4 D10.Dic6 D10.2Dic6
Matrix representation of C12.53D4 ►in GL4(𝔽5) generated by
| 3 | 3 | 0 | 0 |
| 1 | 4 | 0 | 0 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 1 | 4 |
| 0 | 2 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 |
| 0 | 0 | 4 | 2 |
| 0 | 0 | 1 | 4 |
| 0 | 0 | 4 | 4 |
| 4 | 1 | 0 | 0 |
| 1 | 1 | 0 | 0 |
G:=sub<GL(4,GF(5))| [3,1,0,0,3,4,0,0,0,0,3,1,0,0,3,4],[0,1,0,0,2,0,0,0,0,0,3,4,0,0,1,2],[0,0,4,1,0,0,1,1,1,4,0,0,4,4,0,0] >;
C12.53D4 in GAP, Magma, Sage, TeX
C_{12}._{53}D_4 % in TeX
G:=Group("C12.53D4"); // GroupNames label
G:=SmallGroup(96,29);
// by ID
G=gap.SmallGroup(96,29);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,121,31,86,297,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^4=a^6,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^6*b^3>;
// generators/relations
Export
Subgroup lattice of C12.53D4 in TeX
Character table of C12.53D4 in TeX