metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.48D4, C22:3Dic6, C23.25D6, (C2xC6):3Q8, C6.8(C2xQ8), C4:Dic3:8C2, (C2xC4).83D6, C6.39(C2xD4), C3:4(C22:Q8), Dic3:C4:2C2, (C2xDic6):6C2, (C22xC4).7S3, C2.9(C2xDic6), C6.15(C4oD4), C4.23(C3:D4), (C22xC12).6C2, (C2xC6).42C23, C2.17(C4oD12), (C2xC12).91C22, C6.D4.4C2, C22.54(C22xS3), (C22xC6).34C22, (C2xDic3).14C22, C2.5(C2xC3: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, C2xC4, C2xC4, Q8, C23, Dic3, C12, C12, C2xC6, C2xC6, C2xC6, C22:C4, C4:C4, C22xC4, C2xQ8, Dic6, C2xDic3, C2xC12, C2xC12, C22xC6, C22:Q8, Dic3:C4, C4:Dic3, C6.D4, C2xDic6, C22xC12, C12.48D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2xD4, C2xQ8, C4oD4, Dic6, C3:D4, C22xS3, C22:Q8, C2xDic6, C4oD12, C2xC3: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 C4oD4 |
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 C4oD4 |
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 C4oD12 |
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 C4oD12 |
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 C4oD12 |
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 C4oD12 |
(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 (C3xD4).31D4 (C2xC6):8Q16 (C3xD4).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 D4xDic6 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 (Q8xDic3):C2 C6.752- 1+4 C6.152- 1+4 S3xC22: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 Q8xC3: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 (C2xC30):Q8 C60.205D4
C12.48D4 is a maximal quotient of
C12:4(C4:C4) (C2xDic6):7C4 (C2xC42).6S3 C23:2Dic6 C24.17D6 C24.18D6 (C2xDic3):Q8 (C2xC12).54D4 (C2xC12).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 (C2xC30):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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