metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2xC4).3D12, (C2xD12).3C4, (C2xC12).15D4, (C2xQ8).25D6, C4.10D4:5S3, (C4xDic3).1C4, (C6xQ8).1C22, C12.10D4:1C2, C6.13(C23:C4), C3:1(C42.C4), C22.14(D6:C4), C12.23D4.1C2, C2.14(C23.6D6), (C2xC4).3(C4xS3), (C2xC12).3(C2xC4), (C2xC4).3(C3:D4), (C2xC6).7(C22:C4), (C3xC4.10D4):11C2, SmallGroup(192,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2xC4).D12
G = < a,b,c,d | a2=b4=1, c12=b2, d2=b, cbc-1=ab=ba, cac-1=dad-1=ab2, bd=db, dcd-1=b-1c11 >
Subgroups: 240 in 64 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, D6, C2xC6, C42, C22:C4, M4(2), C2xD4, C2xQ8, C3:C8, C24, D12, C2xDic3, C2xC12, C3xQ8, C22xS3, C4.10D4, C4.10D4, C4.4D4, C4.Dic3, C4xDic3, D6:C4, C3xM4(2), C2xD12, C6xQ8, C42.C4, C12.10D4, C3xC4.10D4, C12.23D4, (C2xC4).D12
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, D6, C22:C4, C4xS3, D12, C3:D4, C23:C4, D6:C4, C42.C4, C23.6D6, (C2xC4).D12
Character table of (C2xC4).D12
class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | |
size | 1 | 1 | 2 | 24 | 2 | 4 | 4 | 4 | 12 | 12 | 2 | 4 | 8 | 8 | 24 | 24 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | |
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 | i | -i | -i | i | -1 | -1 | -1 | 1 | -i | -i | i | i | linear of order 4 |
ρ6 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -1 | 1 | -i | -i | i | i | linear of order 4 |
ρ7 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | -1 | -1 | -1 | 1 | i | i | -i | -i | linear of order 4 |
ρ8 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -1 | -1 | 1 | i | i | -i | -i | linear of order 4 |
ρ9 | 2 | 2 | 2 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ11 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
ρ12 | 2 | 2 | 2 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ13 | 2 | 2 | 2 | 0 | -1 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
ρ14 | 2 | 2 | 2 | 0 | -1 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
ρ15 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | -2i | 2i | 0 | 0 | 1 | 1 | 1 | -1 | -i | -i | i | i | complex lifted from C4xS3 |
ρ16 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | 2i | -2i | 0 | 0 | 1 | 1 | 1 | -1 | i | i | -i | -i | complex lifted from C4xS3 |
ρ17 | 2 | 2 | 2 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -√-3 | √-3 | √-3 | -√-3 | complex lifted from C3:D4 |
ρ18 | 2 | 2 | 2 | 0 | -1 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | √-3 | -√-3 | -√-3 | √-3 | complex lifted from C3:D4 |
ρ19 | 4 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23:C4 |
ρ20 | 4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2√-3 | -2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.6D6 |
ρ21 | 4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -2√-3 | 2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C23.6D6 |
ρ22 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | -2i | 2i | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42.C4 |
ρ23 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 2i | -2i | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42.C4 |
ρ24 | 8 | -8 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
(2 14)(4 16)(6 18)(8 20)(10 22)(12 24)(26 38)(28 40)(30 42)(32 44)(34 46)(36 48)
(1 45 13 33)(2 46 14 34)(3 35 15 47)(4 36 16 48)(5 25 17 37)(6 26 18 38)(7 39 19 27)(8 40 20 28)(9 29 21 41)(10 30 22 42)(11 43 23 31)(12 44 24 32)
(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 32 45 12 13 44 33 24)(2 11 46 43 14 23 34 31)(3 30 35 22 15 42 47 10)(4 21 36 41 16 9 48 29)(5 28 25 8 17 40 37 20)(6 7 26 39 18 19 38 27)
G:=sub<Sym(48)| (2,14)(4,16)(6,18)(8,20)(10,22)(12,24)(26,38)(28,40)(30,42)(32,44)(34,46)(36,48), (1,45,13,33)(2,46,14,34)(3,35,15,47)(4,36,16,48)(5,25,17,37)(6,26,18,38)(7,39,19,27)(8,40,20,28)(9,29,21,41)(10,30,22,42)(11,43,23,31)(12,44,24,32), (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), (1,32,45,12,13,44,33,24)(2,11,46,43,14,23,34,31)(3,30,35,22,15,42,47,10)(4,21,36,41,16,9,48,29)(5,28,25,8,17,40,37,20)(6,7,26,39,18,19,38,27)>;
G:=Group( (2,14)(4,16)(6,18)(8,20)(10,22)(12,24)(26,38)(28,40)(30,42)(32,44)(34,46)(36,48), (1,45,13,33)(2,46,14,34)(3,35,15,47)(4,36,16,48)(5,25,17,37)(6,26,18,38)(7,39,19,27)(8,40,20,28)(9,29,21,41)(10,30,22,42)(11,43,23,31)(12,44,24,32), (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), (1,32,45,12,13,44,33,24)(2,11,46,43,14,23,34,31)(3,30,35,22,15,42,47,10)(4,21,36,41,16,9,48,29)(5,28,25,8,17,40,37,20)(6,7,26,39,18,19,38,27) );
G=PermutationGroup([[(2,14),(4,16),(6,18),(8,20),(10,22),(12,24),(26,38),(28,40),(30,42),(32,44),(34,46),(36,48)], [(1,45,13,33),(2,46,14,34),(3,35,15,47),(4,36,16,48),(5,25,17,37),(6,26,18,38),(7,39,19,27),(8,40,20,28),(9,29,21,41),(10,30,22,42),(11,43,23,31),(12,44,24,32)], [(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)], [(1,32,45,12,13,44,33,24),(2,11,46,43,14,23,34,31),(3,30,35,22,15,42,47,10),(4,21,36,41,16,9,48,29),(5,28,25,8,17,40,37,20),(6,7,26,39,18,19,38,27)]])
Matrix representation of (C2xC4).D12 ►in GL6(F73)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 | 72 | 0 |
0 | 0 | 1 | 0 | 0 | 72 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 46 | 54 | 0 | 0 |
0 | 0 | 0 | 27 | 0 | 0 |
0 | 0 | 46 | 27 | 0 | 27 |
0 | 0 | 46 | 27 | 27 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
72 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 71 | 0 |
0 | 0 | 0 | 0 | 72 | 1 |
0 | 0 | 60 | 0 | 72 | 0 |
0 | 0 | 14 | 46 | 72 | 0 |
72 | 1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 71 | 0 |
0 | 0 | 1 | 0 | 72 | 72 |
0 | 0 | 14 | 46 | 72 | 0 |
0 | 0 | 60 | 0 | 72 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,1,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,46,46,0,0,54,27,27,27,0,0,0,0,0,27,0,0,0,0,27,0],[0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,0,60,14,0,0,0,0,0,46,0,0,71,72,72,72,0,0,0,1,0,0],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,14,60,0,0,0,0,46,0,0,0,71,72,72,72,0,0,0,72,0,0] >;
(C2xC4).D12 in GAP, Magma, Sage, TeX
(C_2\times C_4).D_{12}
% in TeX
G:=Group("(C2xC4).D12");
// GroupNames label
G:=SmallGroup(192,36);
// by ID
G=gap.SmallGroup(192,36);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,422,184,1123,794,297,136,851,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=1,c^12=b^2,d^2=b,c*b*c^-1=a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*d=d*b,d*c*d^-1=b^-1*c^11>;
// generators/relations
Export