metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.48D4, C22⋊3Dic6, C23.25D6, (C2×C6)⋊3Q8, C6.8(C2×Q8), C4⋊Dic3⋊8C2, (C2×C4).83D6, C6.39(C2×D4), C3⋊4(C22⋊Q8), Dic3⋊C4⋊2C2, (C2×Dic6)⋊6C2, (C22×C4).7S3, C2.9(C2×Dic6), C6.15(C4○D4), C4.23(C3⋊D4), (C22×C12).6C2, (C2×C6).42C23, C2.17(C4○D12), (C2×C12).91C22, C6.D4.4C2, C22.54(C22×S3), (C22×C6).34C22, (C2×Dic3).14C22, C2.5(C2×C3⋊D4), SmallGroup(96,131)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.48D4
G = < a,b,c | a12=b4=1, c2=a6, bab-1=cac-1=a-1, cbc-1=a6b-1 >
Subgroups: 146 in 74 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, C22⋊Q8, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×Dic6, C22×C12, C12.48D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C3⋊D4, C22×S3, C22⋊Q8, C2×Dic6, C4○D12, C2×C3⋊D4, C12.48D4
Character table of C12.48D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | 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 | -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 | -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 | -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 | -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 | 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 | 1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | -1 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 2 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | -2 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ16 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -√3 | √3 | -√3 | √3 | -√3 | -√3 | √3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ18 | 2 | -2 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -√3 | -√3 | -√3 | √3 | √3 | √3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ19 | 2 | -2 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | √3 | -√3 | √3 | -√3 | √3 | √3 | -√3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ20 | 2 | -2 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | √3 | √3 | √3 | -√3 | -√3 | -√3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | √-3 | -√-3 | -1 | √-3 | -√-3 | √-3 | -1 | -√-3 | √-3 | -1 | 1 | -√-3 | 1 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -√-3 | √-3 | -1 | -√-3 | √-3 | √-3 | 1 | -√-3 | √-3 | 1 | -1 | -√-3 | -1 | complex lifted from C3⋊D4 |
| ρ23 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -√-3 | √-3 | -1 | -√-3 | √-3 | -√-3 | -1 | √-3 | -√-3 | -1 | 1 | √-3 | 1 | complex lifted from C3⋊D4 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | √-3 | -√-3 | -1 | √-3 | -√-3 | -√-3 | 1 | √-3 | -√-3 | 1 | -1 | √-3 | -1 | complex lifted from C3⋊D4 |
| ρ25 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | -2i | 0 | 2i | 2i | 0 | 0 | -2i | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 2i | 0 | -2i | -2i | 0 | 0 | 2i | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -1 | 1 | √-3 | -√-3 | 1 | -√-3 | √-3 | i | -√3 | -i | -i | √3 | -√3 | i | √3 | complex lifted from C4○D12 |
| ρ28 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -1 | 1 | -√-3 | √-3 | 1 | √-3 | -√-3 | i | √3 | -i | -i | -√3 | √3 | i | -√3 | complex lifted from C4○D12 |
| ρ29 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -1 | 1 | √-3 | -√-3 | 1 | -√-3 | √-3 | -i | √3 | i | i | -√3 | √3 | -i | -√3 | complex lifted from C4○D12 |
| ρ30 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -1 | 1 | -√-3 | √-3 | 1 | √-3 | -√-3 | -i | -√3 | i | i | √3 | -√3 | -i | √3 | complex lifted from C4○D12 |
►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 17 32 42)(2 16 33 41)(3 15 34 40)(4 14 35 39)(5 13 36 38)(6 24 25 37)(7 23 26 48)(8 22 27 47)(9 21 28 46)(10 20 29 45)(11 19 30 44)(12 18 31 43)
(1 48 7 42)(2 47 8 41)(3 46 9 40)(4 45 10 39)(5 44 11 38)(6 43 12 37)(13 36 19 30)(14 35 20 29)(15 34 21 28)(16 33 22 27)(17 32 23 26)(18 31 24 25)
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), (1,17,32,42)(2,16,33,41)(3,15,34,40)(4,14,35,39)(5,13,36,38)(6,24,25,37)(7,23,26,48)(8,22,27,47)(9,21,28,46)(10,20,29,45)(11,19,30,44)(12,18,31,43), (1,48,7,42)(2,47,8,41)(3,46,9,40)(4,45,10,39)(5,44,11,38)(6,43,12,37)(13,36,19,30)(14,35,20,29)(15,34,21,28)(16,33,22,27)(17,32,23,26)(18,31,24,25)>;
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), (1,17,32,42)(2,16,33,41)(3,15,34,40)(4,14,35,39)(5,13,36,38)(6,24,25,37)(7,23,26,48)(8,22,27,47)(9,21,28,46)(10,20,29,45)(11,19,30,44)(12,18,31,43), (1,48,7,42)(2,47,8,41)(3,46,9,40)(4,45,10,39)(5,44,11,38)(6,43,12,37)(13,36,19,30)(14,35,20,29)(15,34,21,28)(16,33,22,27)(17,32,23,26)(18,31,24,25) );
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)], [(1,17,32,42),(2,16,33,41),(3,15,34,40),(4,14,35,39),(5,13,36,38),(6,24,25,37),(7,23,26,48),(8,22,27,47),(9,21,28,46),(10,20,29,45),(11,19,30,44),(12,18,31,43)], [(1,48,7,42),(2,47,8,41),(3,46,9,40),(4,45,10,39),(5,44,11,38),(6,43,12,37),(13,36,19,30),(14,35,20,29),(15,34,21,28),(16,33,22,27),(17,32,23,26),(18,31,24,25)]])
C12.48D4 is a maximal subgroup of
C4⋊Dic3⋊C4 C23.35D12 C23.39D12 C23.40D12 C23.15D12 D12.31D4 D12.32D4 Dic6.32D4 C4⋊C4.230D6 C4⋊C4.231D6 C4⋊C4.233D6 C3⋊C8⋊23D4 C3⋊C8⋊5D4 C3⋊C8.29D4 C3⋊C8.6D4 C24⋊30D4 C24.82D4 C24⋊2D4 C24.4D4 (C3×D4).31D4 (C2×C6)⋊8Q16 (C3×D4).32D4 C42.274D6 C42.277D6 C23⋊3Dic6 C24.41D6 C24.42D6 C6.72+ 1+4 C6.102+ 1+4 C6.52- 1+4 C6.62- 1+4 C42.89D6 C42.94D6 C42.98D6 C42.99D6 D4×Dic6 D4⋊5Dic6 C42.105D6 C42.106D6 D4⋊6Dic6 D12⋊23D4 D12⋊24D4 Dic6⋊23D4 C42⋊18D6 C42.115D6 C42.118D6 C4⋊C4.178D6 C6.702- 1+4 C6.712- 1+4 C4⋊C4⋊21D6 C6.722- 1+4 C6.462+ 1+4 C6.492+ 1+4 (Q8×Dic3)⋊C2 C6.752- 1+4 C6.152- 1+4 S3×C22⋊Q8 C6.162- 1+4 C6.512+ 1+4 C6.252- 1+4 C6.812- 1+4 C6.632+ 1+4 C6.652+ 1+4 C6.692+ 1+4 C24.83D6 C24.53D6 Q8×C3⋊D4 C6.1042- 1+4 C6.1052- 1+4 C6.1072- 1+4 C6.1082- 1+4 C36.49D4 C12.1S4 D6⋊Dic6 D6⋊6Dic6 D6⋊7Dic6 C62⋊3Q8 C62⋊10Q8 A4⋊Dic6 D10⋊Dic6 C60.67D4 C60.68D4 (C2×C30)⋊Q8 C60.205D4
C12.48D4 is a maximal quotient of
C12⋊4(C4⋊C4) (C2×Dic6)⋊7C4 (C2×C42).6S3 C23⋊2Dic6 C24.17D6 C24.18D6 (C2×Dic3)⋊Q8 (C2×C12).54D4 (C2×C12).55D4 C12.50D8 C12.38SD16 D4.3Dic6 Q8⋊4Dic6 Q8⋊5Dic6 Q8.5Dic6 C24.73D6 C24.75D6 C36.49D4 D6⋊Dic6 D6⋊6Dic6 D6⋊7Dic6 C62⋊3Q8 C62⋊10Q8 D10⋊Dic6 C60.67D4 C60.68D4 (C2×C30)⋊Q8 C60.205D4
Matrix representation of C12.48D4 ►in GL4(𝔽13) generated by
| 6 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 |
| 0 | 0 | 6 | 3 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 |
| 0 | 0 | 6 | 3 |
G:=sub<GL(4,GF(13))| [6,0,0,0,0,11,0,0,0,0,5,0,0,0,5,8],[0,1,0,0,1,0,0,0,0,0,10,6,0,0,7,3],[0,12,0,0,1,0,0,0,0,0,10,6,0,0,7,3] >;
C12.48D4 in GAP, Magma, Sage, TeX
C_{12}._{48}D_4 % in TeX
G:=Group("C12.48D4"); // GroupNames label
G:=SmallGroup(96,131);
// by ID
G=gap.SmallGroup(96,131);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,217,103,218,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=1,c^2=a^6,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^6*b^-1>;
// generators/relations
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