metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.47D4, C4.12D12, M4(2).2S3, (C2×C4).2D6, (C2×Dic3).C4, C22.5(C4×S3), C2.10(D6⋊C4), C4.22(C3⋊D4), C3⋊1(C4.10D4), C6.9(C22⋊C4), (C2×Dic6).6C2, C4.Dic3.3C2, (C2×C12).14C22, (C3×M4(2)).2C2, (C2×C6).3(C2×C4), SmallGroup(96,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.47D4
G = < a,b,c | a12=1, b4=c2=a6, bab-1=cac-1=a-1, cbc-1=a9b3 >
Character table of C12.47D4
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 4 | 4 | 4 | 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 | 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 | -i | i | i | -i | -1 | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | i | -i | -i | i | -1 | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ9 | 2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ14 | 2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ15 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 2i | -2i | 0 | 0 | 1 | 1 | 1 | -i | -i | i | i | complex lifted from C4×S3 |
| ρ16 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -2i | 2i | 0 | 0 | 1 | 1 | 1 | i | i | -i | -i | complex lifted from C4×S3 |
| ρ17 | 2 | 2 | -2 | -1 | 2 | -2 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -√-3 | √-3 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | -1 | 2 | -2 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | √-3 | -√-3 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ19 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.10D4, Schur index 2 |
| ρ20 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2√3 | 2√3 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2√3 | -2√3 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►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 46 10 37 7 40 4 43)(2 45 11 48 8 39 5 42)(3 44 12 47 9 38 6 41)(13 33 16 30 19 27 22 36)(14 32 17 29 20 26 23 35)(15 31 18 28 21 25 24 34)
(1 25 7 31)(2 36 8 30)(3 35 9 29)(4 34 10 28)(5 33 11 27)(6 32 12 26)(13 45 19 39)(14 44 20 38)(15 43 21 37)(16 42 22 48)(17 41 23 47)(18 40 24 46)
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), (1,46,10,37,7,40,4,43)(2,45,11,48,8,39,5,42)(3,44,12,47,9,38,6,41)(13,33,16,30,19,27,22,36)(14,32,17,29,20,26,23,35)(15,31,18,28,21,25,24,34), (1,25,7,31)(2,36,8,30)(3,35,9,29)(4,34,10,28)(5,33,11,27)(6,32,12,26)(13,45,19,39)(14,44,20,38)(15,43,21,37)(16,42,22,48)(17,41,23,47)(18,40,24,46)>;
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), (1,46,10,37,7,40,4,43)(2,45,11,48,8,39,5,42)(3,44,12,47,9,38,6,41)(13,33,16,30,19,27,22,36)(14,32,17,29,20,26,23,35)(15,31,18,28,21,25,24,34), (1,25,7,31)(2,36,8,30)(3,35,9,29)(4,34,10,28)(5,33,11,27)(6,32,12,26)(13,45,19,39)(14,44,20,38)(15,43,21,37)(16,42,22,48)(17,41,23,47)(18,40,24,46) );
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)], [(1,46,10,37,7,40,4,43),(2,45,11,48,8,39,5,42),(3,44,12,47,9,38,6,41),(13,33,16,30,19,27,22,36),(14,32,17,29,20,26,23,35),(15,31,18,28,21,25,24,34)], [(1,25,7,31),(2,36,8,30),(3,35,9,29),(4,34,10,28),(5,33,11,27),(6,32,12,26),(13,45,19,39),(14,44,20,38),(15,43,21,37),(16,42,22,48),(17,41,23,47),(18,40,24,46)]])
C12.47D4 is a maximal subgroup of
M4(2).19D6 D12.2D4 S3×C4.10D4 D12.7D4 Q8.14D12 D4.10D12 C24.18D4 C24.42D4 M4(2).31D6 Q8.8D12 Q8.10D12 M4(2).13D6 D12.38D4 M4(2).16D6 D12.40D4 C4.D36 C12.14D12 C12.71D12 C12.20D12 C60.54D4 C60.31D4 C4.D60 Dic5.4D12
C12.47D4 is a maximal quotient of
C42.2D6 (C2×Dic3)⋊C8 C12.47D8 C12.2D8 M4(2)⋊Dic3 C4.D36 C12.14D12 C12.71D12 C12.20D12 C60.54D4 C60.31D4 C4.D60 Dic5.4D12
Matrix representation of C12.47D4 ►in GL4(𝔽73) generated by
| 66 | 66 | 0 | 0 |
| 7 | 59 | 0 | 0 |
| 0 | 0 | 66 | 66 |
| 0 | 0 | 7 | 59 |
| 0 | 0 | 71 | 20 |
| 0 | 0 | 18 | 2 |
| 47 | 34 | 0 | 0 |
| 8 | 26 | 0 | 0 |
| 71 | 20 | 0 | 0 |
| 18 | 2 | 0 | 0 |
| 0 | 0 | 71 | 20 |
| 0 | 0 | 18 | 2 |
G:=sub<GL(4,GF(73))| [66,7,0,0,66,59,0,0,0,0,66,7,0,0,66,59],[0,0,47,8,0,0,34,26,71,18,0,0,20,2,0,0],[71,18,0,0,20,2,0,0,0,0,71,18,0,0,20,2] >;
C12.47D4 in GAP, Magma, Sage, TeX
C_{12}._{47}D_4 % in TeX
G:=Group("C12.47D4"); // GroupNames label
G:=SmallGroup(96,31);
// by ID
G=gap.SmallGroup(96,31);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,121,31,362,86,297,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^4=c^2=a^6,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^9*b^3>;
// generators/relations
Export
Subgroup lattice of C12.47D4 in TeX
Character table of C12.47D4 in TeX