metabelian, supersoluble, monomial
Aliases: C32:7D4, C62:3C2, C6.16D6, (C2xC6):4S3, C3:3(C3:D4), C3:Dic3:3C2, C22:2(C3:S3), (C3xC6).15C22, (C2xC3:S3):3C2, C2.5(C2xC3:S3), SmallGroup(72,35)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32:7D4
G = < a,b,c,d | a3=b3=c4=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >
Quotients: C1, C2, C22, S3, D4, D6, C3:S3, C3:D4, C2xC3:S3, C32:7D4
Character table of C32:7D4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | |
| size | 1 | 1 | 2 | 18 | 2 | 2 | 2 | 2 | 18 | 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 | trivial |
| ρ2 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | 0 | -1 | -1 | -1 | 2 | 0 | -1 | -1 | 1 | 1 | 1 | -2 | 1 | 1 | 1 | -2 | -1 | 2 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | -2 | 0 | -1 | 2 | -1 | -1 | 0 | -1 | 2 | -2 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | -2 | 0 | -1 | -1 | 2 | -1 | 0 | -1 | -1 | 1 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | 2 | -1 | orthogonal lifted from D6 |
| ρ9 | 2 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | 0 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | -1 | -1 | -1 | 0 | 2 | -1 | 1 | 1 | 1 | 1 | -2 | 1 | -2 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | 0 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 0 | 1 | 1 | -√-3 | √-3 | 0 | -√-3 | √-3 | 0 | -√-3 | √-3 | -2 | 1 | complex lifted from C3:D4 |
| ρ15 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 0 | 1 | 1 | √-3 | -√-3 | 0 | √-3 | -√-3 | 0 | √-3 | -√-3 | -2 | 1 | complex lifted from C3:D4 |
| ρ16 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 0 | 1 | -2 | 0 | 0 | √-3 | √-3 | √-3 | -√-3 | -√-3 | -√-3 | 1 | 1 | complex lifted from C3:D4 |
| ρ17 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 0 | 1 | -2 | 0 | 0 | -√-3 | -√-3 | -√-3 | √-3 | √-3 | √-3 | 1 | 1 | complex lifted from C3:D4 |
| ρ18 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | 1 | 1 | -√-3 | √-3 | √-3 | 0 | -√-3 | -√-3 | √-3 | 0 | 1 | -2 | complex lifted from C3:D4 |
| ρ19 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | 1 | 1 | √-3 | -√-3 | -√-3 | 0 | √-3 | √-3 | -√-3 | 0 | 1 | -2 | complex lifted from C3:D4 |
| ρ20 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | -2 | 1 | -√-3 | √-3 | -√-3 | √-3 | 0 | √-3 | 0 | -√-3 | 1 | 1 | complex lifted from C3:D4 |
| ρ21 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | -2 | 1 | √-3 | -√-3 | √-3 | -√-3 | 0 | -√-3 | 0 | √-3 | 1 | 1 | complex lifted from C3:D4 |
►On 36 points
Generators in S36
(1 16 11)(2 12 13)(3 14 9)(4 10 15)(5 31 26)(6 27 32)(7 29 28)(8 25 30)(17 36 22)(18 23 33)(19 34 24)(20 21 35)
(1 8 21)(2 22 5)(3 6 23)(4 24 7)(9 32 18)(10 19 29)(11 30 20)(12 17 31)(13 36 26)(14 27 33)(15 34 28)(16 25 35)
(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)
(2 4)(5 24)(6 23)(7 22)(8 21)(9 14)(10 13)(11 16)(12 15)(17 28)(18 27)(19 26)(20 25)(29 36)(30 35)(31 34)(32 33)
G:=sub<Sym(36)| (1,16,11)(2,12,13)(3,14,9)(4,10,15)(5,31,26)(6,27,32)(7,29,28)(8,25,30)(17,36,22)(18,23,33)(19,34,24)(20,21,35), (1,8,21)(2,22,5)(3,6,23)(4,24,7)(9,32,18)(10,19,29)(11,30,20)(12,17,31)(13,36,26)(14,27,33)(15,34,28)(16,25,35), (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), (2,4)(5,24)(6,23)(7,22)(8,21)(9,14)(10,13)(11,16)(12,15)(17,28)(18,27)(19,26)(20,25)(29,36)(30,35)(31,34)(32,33)>;
G:=Group( (1,16,11)(2,12,13)(3,14,9)(4,10,15)(5,31,26)(6,27,32)(7,29,28)(8,25,30)(17,36,22)(18,23,33)(19,34,24)(20,21,35), (1,8,21)(2,22,5)(3,6,23)(4,24,7)(9,32,18)(10,19,29)(11,30,20)(12,17,31)(13,36,26)(14,27,33)(15,34,28)(16,25,35), (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), (2,4)(5,24)(6,23)(7,22)(8,21)(9,14)(10,13)(11,16)(12,15)(17,28)(18,27)(19,26)(20,25)(29,36)(30,35)(31,34)(32,33) );
G=PermutationGroup([[(1,16,11),(2,12,13),(3,14,9),(4,10,15),(5,31,26),(6,27,32),(7,29,28),(8,25,30),(17,36,22),(18,23,33),(19,34,24),(20,21,35)], [(1,8,21),(2,22,5),(3,6,23),(4,24,7),(9,32,18),(10,19,29),(11,30,20),(12,17,31),(13,36,26),(14,27,33),(15,34,28),(16,25,35)], [(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)], [(2,4),(5,24),(6,23),(7,22),(8,21),(9,14),(10,13),(11,16),(12,15),(17,28),(18,27),(19,26),(20,25),(29,36),(30,35),(31,34),(32,33)]])
C32:7D4 is a maximal subgroup of
D6.3D6 S3xC3:D4 C12.59D6 D4xC3:S3 C12.D6 He3:6D4 C6.D18 C32.3S4 C33:6D4 C33:7D4 C33:15D4 C32:4S4 SL2(F3).D6 (C2xC6):4S4 C30.12D6 C32:7D20 C62:D5
C32:7D4 is a maximal quotient of
C6.Dic6 C6.11D12 C32:7D8 C32:9SD16 C32:11SD16 C32:7Q16 C62:5C4 C6.D18 He3:7D4 C33:6D4 C33:7D4 C33:15D4 (C2xC6):4S4 C30.12D6 C32:7D20 C62:D5
Matrix representation of C32:7D4 ►in GL4(F13) generated by
| 0 | 1 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 11 | 2 |
| 0 | 0 | 4 | 2 |
| 1 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [0,12,0,0,1,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,12,0,0,1,12],[1,12,0,0,0,12,0,0,0,0,11,4,0,0,2,2],[1,12,0,0,0,12,0,0,0,0,1,0,0,0,1,12] >;
C32:7D4 in GAP, Magma, Sage, TeX
C_3^2\rtimes_7D_4
% in TeX
G:=Group("C3^2:7D4"); // GroupNames label
G:=SmallGroup(72,35);
// by ID
G=gap.SmallGroup(72,35);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,61,323,1204]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C32:7D4 in TeX
Character table of C32:7D4 in TeX