metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.10D4, (C2×C4).4D6, (C2×C4).Dic3, (C2×C12).1C4, (C6×Q8).2C2, (C2×Q8).4S3, C4.15(C3⋊D4), C3⋊2(C4.10D4), C4.Dic3.4C2, C6.17(C22⋊C4), (C2×C12).19C22, C22.4(C2×Dic3), C2.7(C6.D4), (C2×C6).30(C2×C4), SmallGroup(96,43)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.10D4
G = < a,b,c | a12=1, b4=a6, c2=a9, bab-1=a-1, cac-1=a5, cbc-1=a9b3 >
Character table of C12.10D4
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 4 | 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 | 1 | -i | i | i | -i | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | i | -i | -i | i | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | i | i | -i | -i | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | -i | i | i | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | -1 | 2 | 2 | -2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ14 | 2 | 2 | 2 | -1 | -2 | -2 | 2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | √-3 | -√-3 | 1 | -√-3 | √-3 | -1 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | -2 | -1 | 2 | -2 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | -√-3 | √-3 | -1 | -√-3 | √-3 | 1 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | -2 | -1 | 2 | -2 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | √-3 | -√-3 | -1 | √-3 | -√-3 | 1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | -√-3 | √-3 | 1 | √-3 | -√-3 | -1 | complex lifted from C3⋊D4 |
| ρ19 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 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√-3 | 2√-3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ21 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2√-3 | -2√-3 | 2 | 0 | 0 | 0 | 0 | 0 | 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 43 4 40 7 37 10 46)(2 42 5 39 8 48 11 45)(3 41 6 38 9 47 12 44)(13 28 16 25 19 34 22 31)(14 27 17 36 20 33 23 30)(15 26 18 35 21 32 24 29)
(1 25 10 34 7 31 4 28)(2 30 11 27 8 36 5 33)(3 35 12 32 9 29 6 26)(13 46 22 43 19 40 16 37)(14 39 23 48 20 45 17 42)(15 44 24 41 21 38 18 47)
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), (1,43,4,40,7,37,10,46)(2,42,5,39,8,48,11,45)(3,41,6,38,9,47,12,44)(13,28,16,25,19,34,22,31)(14,27,17,36,20,33,23,30)(15,26,18,35,21,32,24,29), (1,25,10,34,7,31,4,28)(2,30,11,27,8,36,5,33)(3,35,12,32,9,29,6,26)(13,46,22,43,19,40,16,37)(14,39,23,48,20,45,17,42)(15,44,24,41,21,38,18,47)>;
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), (1,43,4,40,7,37,10,46)(2,42,5,39,8,48,11,45)(3,41,6,38,9,47,12,44)(13,28,16,25,19,34,22,31)(14,27,17,36,20,33,23,30)(15,26,18,35,21,32,24,29), (1,25,10,34,7,31,4,28)(2,30,11,27,8,36,5,33)(3,35,12,32,9,29,6,26)(13,46,22,43,19,40,16,37)(14,39,23,48,20,45,17,42)(15,44,24,41,21,38,18,47) );
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)], [(1,43,4,40,7,37,10,46),(2,42,5,39,8,48,11,45),(3,41,6,38,9,47,12,44),(13,28,16,25,19,34,22,31),(14,27,17,36,20,33,23,30),(15,26,18,35,21,32,24,29)], [(1,25,10,34,7,31,4,28),(2,30,11,27,8,36,5,33),(3,35,12,32,9,29,6,26),(13,46,22,43,19,40,16,37),(14,39,23,48,20,45,17,42),(15,44,24,41,21,38,18,47)]])
C12.10D4 is a maximal subgroup of
(C2×C4).D12 (C2×C12).D4 C42.Dic3 C42.3Dic3 S3×C4.10D4 M4(2).21D6 D12.14D4 D12.15D4 C24.44D4 C24.29D4 M4(2).15D6 M4(2).16D6 (C6×D4).16C4 2- 1+4⋊4S3 2- 1+4.2S3 C36.9D4 C12.14D12 (C6×C12).C4 C12.6D20 C60.10D4 (C2×C60).C4
C12.10D4 is a maximal quotient of
(C2×C12)⋊C8 C12.(C4⋊C4) C42.8D6 C12.10D8 C36.9D4 C12.14D12 (C6×C12).C4 C12.6D20 C60.10D4 (C2×C60).C4
Matrix representation of C12.10D4 ►in GL4(𝔽7) generated by
| 0 | 3 | 1 | 0 |
| 0 | 3 | 0 | 4 |
| 2 | 6 | 0 | 6 |
| 6 | 0 | 3 | 4 |
| 5 | 4 | 4 | 1 |
| 5 | 6 | 0 | 6 |
| 2 | 3 | 5 | 1 |
| 1 | 5 | 1 | 5 |
| 5 | 3 | 6 | 3 |
| 0 | 3 | 5 | 4 |
| 1 | 1 | 0 | 2 |
| 5 | 1 | 6 | 6 |
G:=sub<GL(4,GF(7))| [0,0,2,6,3,3,6,0,1,0,0,3,0,4,6,4],[5,5,2,1,4,6,3,5,4,0,5,1,1,6,1,5],[5,0,1,5,3,3,1,1,6,5,0,6,3,4,2,6] >;
C12.10D4 in GAP, Magma, Sage, TeX
C_{12}._{10}D_4 % in TeX
G:=Group("C12.10D4"); // GroupNames label
G:=SmallGroup(96,43);
// by ID
G=gap.SmallGroup(96,43);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,103,188,86,579,2309]);
// 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^9*b^3>;
// generators/relations
Export
Subgroup lattice of C12.10D4 in TeX
Character table of C12.10D4 in TeX