metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊2Q8, C12.11D4, C4.13D12, C4⋊C4⋊5S3, C6.8(C2×D4), C2.7(S3×Q8), D6⋊C4.3C2, C4⋊Dic3⋊6C2, (C2×C4).14D6, C3⋊3(C22⋊Q8), C6.14(C2×Q8), (C2×Dic6)⋊7C2, C2.10(C2×D12), C6.27(C4○D4), (C2×C6).38C23, (C2×C12).6C22, C2.13(D4⋊2S3), C22.52(C22×S3), (C22×S3).22C22, (C2×Dic3).13C22, (C3×C4⋊C4)⋊8C2, (S3×C2×C4).3C2, SmallGroup(96,104)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.D12
G = < a,b,c | a4=b12=1, c2=a2, bab-1=cac-1=a-1, cbc-1=a2b-1 >
Subgroups: 170 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22⋊Q8, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C4.D12
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, D12, C22×S3, C22⋊Q8, C2×D12, D4⋊2S3, S3×Q8, C4.D12
Character table of C4.D12
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 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 | 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 | -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 | 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 | -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 | -1 | 1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | √3 | √3 | 1 | -√3 | -√3 | orthogonal lifted from D12 |
| ρ15 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | √3 | -√3 | -1 | √3 | -√3 | orthogonal lifted from D12 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -√3 | √3 | -1 | -√3 | √3 | orthogonal lifted from D12 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | -√3 | -√3 | 1 | √3 | √3 | orthogonal lifted from D12 |
| ρ19 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ20 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from S3×Q8, Schur index 2 |
►On 48 points
(1 44 25 24)(2 13 26 45)(3 46 27 14)(4 15 28 47)(5 48 29 16)(6 17 30 37)(7 38 31 18)(8 19 32 39)(9 40 33 20)(10 21 34 41)(11 42 35 22)(12 23 36 43)
(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 36 25 12)(2 11 26 35)(3 34 27 10)(4 9 28 33)(5 32 29 8)(6 7 30 31)(13 22 45 42)(14 41 46 21)(15 20 47 40)(16 39 48 19)(17 18 37 38)(23 24 43 44)
G:=sub<Sym(48)| (1,44,25,24)(2,13,26,45)(3,46,27,14)(4,15,28,47)(5,48,29,16)(6,17,30,37)(7,38,31,18)(8,19,32,39)(9,40,33,20)(10,21,34,41)(11,42,35,22)(12,23,36,43), (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), (1,36,25,12)(2,11,26,35)(3,34,27,10)(4,9,28,33)(5,32,29,8)(6,7,30,31)(13,22,45,42)(14,41,46,21)(15,20,47,40)(16,39,48,19)(17,18,37,38)(23,24,43,44)>;
G:=Group( (1,44,25,24)(2,13,26,45)(3,46,27,14)(4,15,28,47)(5,48,29,16)(6,17,30,37)(7,38,31,18)(8,19,32,39)(9,40,33,20)(10,21,34,41)(11,42,35,22)(12,23,36,43), (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), (1,36,25,12)(2,11,26,35)(3,34,27,10)(4,9,28,33)(5,32,29,8)(6,7,30,31)(13,22,45,42)(14,41,46,21)(15,20,47,40)(16,39,48,19)(17,18,37,38)(23,24,43,44) );
G=PermutationGroup([[(1,44,25,24),(2,13,26,45),(3,46,27,14),(4,15,28,47),(5,48,29,16),(6,17,30,37),(7,38,31,18),(8,19,32,39),(9,40,33,20),(10,21,34,41),(11,42,35,22),(12,23,36,43)], [(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)], [(1,36,25,12),(2,11,26,35),(3,34,27,10),(4,9,28,33),(5,32,29,8),(6,7,30,31),(13,22,45,42),(14,41,46,21),(15,20,47,40),(16,39,48,19),(17,18,37,38),(23,24,43,44)]])
C4.D12 is a maximal subgroup of
D6⋊5SD16 D6⋊SD16 C3⋊C8⋊1D4 D4.D12 Q8.11D12 D6⋊Q16 D6⋊1Q16 C3⋊C8.D4 D6.2SD16 C8⋊8D12 C8.2D12 C6.(C4○D8) D6.2Q16 C8⋊3D12 D6⋊2Q16 C2.D8⋊7S3 C6.2+ 1+4 C6.102+ 1+4 C6.52- 1+4 C42⋊10D6 C42.92D6 C42.94D6 C42.98D6 D4⋊5D12 D4⋊6D12 C42.229D6 C42.115D6 Q8×D12 Q8⋊6D12 C42.232D6 C42.134D6 Dic6⋊19D4 C4⋊C4⋊21D6 C6.722- 1+4 D12⋊20D4 S3×C22⋊Q8 C6.162- 1+4 D12⋊22D4 Dic6⋊21D4 C6.512+ 1+4 C6.1182+ 1+4 C6.242- 1+4 C6.592+ 1+4 C6.612+ 1+4 C6.1222+ 1+4 C6.852- 1+4 C6.692+ 1+4 C42.148D6 D12⋊7Q8 C42.237D6 C42.150D6 C42.152D6 C42.156D6 C42.157D6 C42⋊26D6 C42.161D6 C42.164D6 C42.165D6 C42.241D6 D12⋊9Q8 C42.177D6 C42.178D6 D18⋊2Q8 D6⋊7Dic6 C12.30D12 D6⋊2Dic6 C62.65C23 C12.31D12 C60.45D4 D30⋊9Q8 D6⋊2Dic10 D30⋊2Q8 D30⋊6Q8
C4.D12 is a maximal quotient of
C2.(C4×D12) (C2×C4)⋊Dic6 (C2×C4).17D12 D6⋊C4⋊C4 (C22×S3)⋊Q8 (C2×C12).33D4 Dic6.3Q8 D12⋊3Q8 D12⋊4Q8 D12.3Q8 Dic6⋊3Q8 Dic6⋊4Q8 C4.(D6⋊C4) (C2×C4).44D12 C4⋊C4⋊6Dic3 C4⋊(D6⋊C4) (C2×C12).56D4 D18⋊2Q8 D6⋊7Dic6 C12.30D12 D6⋊2Dic6 C62.65C23 C12.31D12 C60.45D4 D30⋊9Q8 D6⋊2Dic10 D30⋊2Q8 D30⋊6Q8
Matrix representation of C4.D12 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 8 |
| 6 | 10 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 3 | 3 | 0 | 0 |
| 6 | 10 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,5,0,0,0,0,8],[6,3,0,0,10,3,0,0,0,0,0,1,0,0,1,0],[3,6,0,0,3,10,0,0,0,0,0,12,0,0,1,0] >;
C4.D12 in GAP, Magma, Sage, TeX
C_4.D_{12} % in TeX
G:=Group("C4.D12"); // GroupNames label
G:=SmallGroup(96,104);
// by ID
G=gap.SmallGroup(96,104);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,218,188,122,2309]);
// Polycyclic
G:=Group<a,b,c|a^4=b^12=1,c^2=a^2,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^2*b^-1>;
// generators/relations
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