non-abelian, soluble, monomial
Aliases: C62.12D4, C32:D8:4C2, C22.5S3wrC2, D6.4D6:1C2, C32:2(C8:C22), C3:Dic3.30D4, D6:S3:2C22, C62.C4:1C2, C32:2C8:2C22, C32:2Q8:2C22, C32:2SD16:5C2, C3:Dic3.8C23, C2.17(C2xS3wrC2), (C3xC6).17(C2xD4), (C2xD6:S3):12C2, (C2xC3:Dic3).95C22, SmallGroup(288,884)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.12D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=a3b, dad=a-1, cbc-1=a2b3, bd=db, dcd=b3c3 >
Subgroups: 592 in 115 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C8, C2xC4, D4, Q8, C23, C32, Dic3, C12, D6, C2xC6, M4(2), D8, SD16, C2xD4, C4oD4, C3xS3, C3xC6, C3xC6, Dic6, C4xS3, C2xDic3, C3:D4, C3xD4, C22xS3, C22xC6, C8:C22, C3xDic3, C3:Dic3, S3xC6, C62, D4:2S3, C2xC3:D4, C32:2C8, S3xDic3, D6:S3, D6:S3, D6:S3, C32:2Q8, C3xC3:D4, C2xC3:Dic3, S3xC2xC6, C32:D8, C32:2SD16, C62.C4, D6.4D6, C2xD6:S3, C62.12D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C8:C22, S3wrC2, C2xS3wrC2, C62.12D4
Character table of C62.12D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 12 | |
| size | 1 | 1 | 2 | 12 | 12 | 12 | 4 | 4 | 12 | 18 | 18 | 4 | 4 | 4 | 4 | 8 | 12 | 12 | 12 | 12 | 24 | 36 | 36 | 24 | |
| 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 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | -4 | 0 | 2 | -2 | -2 | 1 | 0 | 0 | 0 | -1 | -1 | 1 | -2 | 2 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS3wrC2 |
| ρ12 | 4 | 4 | 4 | 0 | 2 | 2 | -2 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | -2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3wrC2 |
| ρ13 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8:C22 |
| ρ14 | 4 | 4 | -4 | 0 | -2 | 2 | -2 | 1 | 0 | 0 | 0 | -1 | -1 | 1 | -2 | 2 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS3wrC2 |
| ρ15 | 4 | 4 | -4 | -2 | 0 | 0 | 1 | -2 | 2 | 0 | 0 | 2 | 2 | -2 | 1 | -1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | -1 | orthogonal lifted from C2xS3wrC2 |
| ρ16 | 4 | 4 | -4 | 2 | 0 | 0 | 1 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 1 | -1 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 1 | orthogonal lifted from C2xS3wrC2 |
| ρ17 | 4 | 4 | 4 | 2 | 0 | 0 | 1 | -2 | 2 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | orthogonal lifted from S3wrC2 |
| ρ18 | 4 | 4 | 4 | -2 | 0 | 0 | 1 | -2 | -2 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | orthogonal lifted from S3wrC2 |
| ρ19 | 4 | 4 | 4 | 0 | -2 | -2 | -2 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | -2 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3wrC2 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 3 | -3 | -1 | 2 | 0 | √-3 | -√-3 | -√-3 | √-3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 3 | -3 | -1 | 2 | 0 | -√-3 | √-3 | √-3 | -√-3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | -3 | 3 | -1 | 2 | 0 | -√-3 | -√-3 | √-3 | √-3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | -3 | 3 | -1 | 2 | 0 | √-3 | √-3 | -√-3 | -√-3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 8 | -8 | 0 | 0 | 0 | 0 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 24 points - transitive group 24T601
(1 5)(2 18 11)(3 7)(4 13 20)(6 22 15)(8 9 24)(10 14)(12 16)(17 21)(19 23)
(1 21 10 5 17 14)(2 6)(3 16 19 7 12 23)(4 8)(9 13)(11 15)(18 22)(20 24)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(18 24)(19 23)(20 22)
G:=sub<Sym(24)| (1,5)(2,18,11)(3,7)(4,13,20)(6,22,15)(8,9,24)(10,14)(12,16)(17,21)(19,23), (1,21,10,5,17,14)(2,6)(3,16,19,7,12,23)(4,8)(9,13)(11,15)(18,22)(20,24), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(18,24)(19,23)(20,22)>;
G:=Group( (1,5)(2,18,11)(3,7)(4,13,20)(6,22,15)(8,9,24)(10,14)(12,16)(17,21)(19,23), (1,21,10,5,17,14)(2,6)(3,16,19,7,12,23)(4,8)(9,13)(11,15)(18,22)(20,24), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(18,24)(19,23)(20,22) );
G=PermutationGroup([[(1,5),(2,18,11),(3,7),(4,13,20),(6,22,15),(8,9,24),(10,14),(12,16),(17,21),(19,23)], [(1,21,10,5,17,14),(2,6),(3,16,19,7,12,23),(4,8),(9,13),(11,15),(18,22),(20,24)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15),(18,24),(19,23),(20,22)]])
G:=TransitiveGroup(24,601);
►On 24 points - transitive group 24T603
(1 14 24 5 10 20)(3 22 12 7 18 16)
(1 5)(2 21 11 6 17 15)(3 7)(4 9 19 8 13 23)(10 14)(12 16)(18 22)(20 24)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 8)(3 7)(4 6)(9 17)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)
G:=sub<Sym(24)| (1,14,24,5,10,20)(3,22,12,7,18,16), (1,5)(2,21,11,6,17,15)(3,7)(4,9,19,8,13,23)(10,14)(12,16)(18,22)(20,24), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,8)(3,7)(4,6)(9,17)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)>;
G:=Group( (1,14,24,5,10,20)(3,22,12,7,18,16), (1,5)(2,21,11,6,17,15)(3,7)(4,9,19,8,13,23)(10,14)(12,16)(18,22)(20,24), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,8)(3,7)(4,6)(9,17)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18) );
G=PermutationGroup([[(1,14,24,5,10,20),(3,22,12,7,18,16)], [(1,5),(2,21,11,6,17,15),(3,7),(4,9,19,8,13,23),(10,14),(12,16),(18,22),(20,24)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,8),(3,7),(4,6),(9,17),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18)]])
G:=TransitiveGroup(24,603);
Matrix representation of C62.12D4 ►in GL4(F7) generated by
| 0 | 1 | 1 | 5 |
| 6 | 5 | 4 | 2 |
| 4 | 4 | 0 | 6 |
| 0 | 0 | 0 | 6 |
| 2 | 4 | 2 | 4 |
| 4 | 2 | 5 | 4 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 3 |
| 6 | 4 | 3 | 5 |
| 6 | 6 | 3 | 0 |
| 2 | 5 | 6 | 6 |
| 2 | 2 | 5 | 3 |
| 1 | 0 | 4 | 5 |
| 0 | 1 | 5 | 5 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
G:=sub<GL(4,GF(7))| [0,6,4,0,1,5,4,0,1,4,0,0,5,2,6,6],[2,4,0,0,4,2,0,0,2,5,6,0,4,4,0,3],[6,6,2,2,4,6,5,2,3,3,6,5,5,0,6,3],[1,0,0,0,0,1,0,0,4,5,6,0,5,5,0,6] >;
C62.12D4 in GAP, Magma, Sage, TeX
C_6^2._{12}D_4 % in TeX
G:=Group("C6^2.12D4"); // GroupNames label
G:=SmallGroup(288,884);
// by ID
G=gap.SmallGroup(288,884);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,219,675,346,80,2693,2028,362,797,1203]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=d^2=1,c^4=b^3,a*b=b*a,c*a*c^-1=a^3*b,d*a*d=a^-1,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d=b^3*c^3>;
// generators/relations
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