metabelian, soluble, monomial, A-group
Aliases: C33⋊2C4, C32⋊3Dic3, C3⋊S3.S3, C3⋊(C32⋊C4), (C3×C3⋊S3).2C2, SmallGroup(108,37)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C33⋊C4 |
Generators and relations for C33⋊C4
G = < a,b,c,d | a3=b3=c3=d4=1, ab=ba, ac=ca, dad-1=ab-1, bc=cb, dbd-1=a-1b-1, dcd-1=c-1 >
Character table of C33⋊C4
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 4A | 4B | 6 | |
| size | 1 | 9 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 27 | 27 | 18 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | i | -i | -1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | i | -1 | linear of order 4 |
| ρ5 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ7 | 4 | 0 | 4 | 1 | -2 | 1 | -2 | 1 | -2 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ8 | 4 | 0 | 4 | -2 | 1 | -2 | 1 | -2 | 1 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ9 | 4 | 0 | -2 | 1 | -1+3√-3/2 | -2 | 1 | 1 | -1-3√-3/2 | 0 | 0 | 0 | complex faithful |
| ρ10 | 4 | 0 | -2 | -1+3√-3/2 | 1 | 1 | -2 | -1-3√-3/2 | 1 | 0 | 0 | 0 | complex faithful |
| ρ11 | 4 | 0 | -2 | 1 | -1-3√-3/2 | -2 | 1 | 1 | -1+3√-3/2 | 0 | 0 | 0 | complex faithful |
| ρ12 | 4 | 0 | -2 | -1-3√-3/2 | 1 | 1 | -2 | -1+3√-3/2 | 1 | 0 | 0 | 0 | complex faithful |
►On 12 points - transitive group 12T72
Generators in S12
(1 8 9)(2 5 10)(3 11 6)(4 12 7)
(2 10 5)(4 7 12)
(1 9 8)(2 5 10)(3 11 6)(4 7 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
G:=sub<Sym(12)| (1,8,9)(2,5,10)(3,11,6)(4,12,7), (2,10,5)(4,7,12), (1,9,8)(2,5,10)(3,11,6)(4,7,12), (1,2,3,4)(5,6,7,8)(9,10,11,12)>;
G:=Group( (1,8,9)(2,5,10)(3,11,6)(4,12,7), (2,10,5)(4,7,12), (1,9,8)(2,5,10)(3,11,6)(4,7,12), (1,2,3,4)(5,6,7,8)(9,10,11,12) );
G=PermutationGroup([[(1,8,9),(2,5,10),(3,11,6),(4,12,7)], [(2,10,5),(4,7,12)], [(1,9,8),(2,5,10),(3,11,6),(4,7,12)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)]])
G:=TransitiveGroup(12,72);
►On 18 points - transitive group 18T54
Generators in S18
(1 17 15)(4 8 10)(6 14 12)
(1 17 15)(2 18 16)(3 9 7)(4 8 10)(5 11 13)(6 14 12)
(1 4 6)(2 5 3)(7 16 13)(8 14 17)(9 18 11)(10 12 15)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
G:=sub<Sym(18)| (1,17,15)(4,8,10)(6,14,12), (1,17,15)(2,18,16)(3,9,7)(4,8,10)(5,11,13)(6,14,12), (1,4,6)(2,5,3)(7,16,13)(8,14,17)(9,18,11)(10,12,15), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18)>;
G:=Group( (1,17,15)(4,8,10)(6,14,12), (1,17,15)(2,18,16)(3,9,7)(4,8,10)(5,11,13)(6,14,12), (1,4,6)(2,5,3)(7,16,13)(8,14,17)(9,18,11)(10,12,15), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18) );
G=PermutationGroup([[(1,17,15),(4,8,10),(6,14,12)], [(1,17,15),(2,18,16),(3,9,7),(4,8,10),(5,11,13),(6,14,12)], [(1,4,6),(2,5,3),(7,16,13),(8,14,17),(9,18,11),(10,12,15)], [(1,2),(3,4),(5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)]])
G:=TransitiveGroup(18,54);
►On 27 points - transitive group 27T31
Generators in S27
(1 6 4)(2 23 21)(3 15 13)(5 18 17)(7 19 16)(8 27 14)(9 26 20)(10 12 25)(11 22 24)
(1 19 17)(2 25 27)(3 9 11)(4 7 18)(5 6 16)(8 21 12)(10 14 23)(13 20 24)(15 26 22)
(1 3 2)(4 13 21)(5 22 14)(6 15 23)(7 20 12)(8 18 24)(9 25 19)(10 16 26)(11 27 17)
(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)
G:=sub<Sym(27)| (1,6,4)(2,23,21)(3,15,13)(5,18,17)(7,19,16)(8,27,14)(9,26,20)(10,12,25)(11,22,24), (1,19,17)(2,25,27)(3,9,11)(4,7,18)(5,6,16)(8,21,12)(10,14,23)(13,20,24)(15,26,22), (1,3,2)(4,13,21)(5,22,14)(6,15,23)(7,20,12)(8,18,24)(9,25,19)(10,16,26)(11,27,17), (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)>;
G:=Group( (1,6,4)(2,23,21)(3,15,13)(5,18,17)(7,19,16)(8,27,14)(9,26,20)(10,12,25)(11,22,24), (1,19,17)(2,25,27)(3,9,11)(4,7,18)(5,6,16)(8,21,12)(10,14,23)(13,20,24)(15,26,22), (1,3,2)(4,13,21)(5,22,14)(6,15,23)(7,20,12)(8,18,24)(9,25,19)(10,16,26)(11,27,17), (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) );
G=PermutationGroup([[(1,6,4),(2,23,21),(3,15,13),(5,18,17),(7,19,16),(8,27,14),(9,26,20),(10,12,25),(11,22,24)], [(1,19,17),(2,25,27),(3,9,11),(4,7,18),(5,6,16),(8,21,12),(10,14,23),(13,20,24),(15,26,22)], [(1,3,2),(4,13,21),(5,22,14),(6,15,23),(7,20,12),(8,18,24),(9,25,19),(10,16,26),(11,27,17)], [(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)]])
G:=TransitiveGroup(27,31);
C33⋊C4 is a maximal subgroup of
S3×C32⋊C4 C33⋊D4 C33⋊Q8 C32⋊3Dic9 C34⋊C4 C62⋊Dic3
C33⋊C4 is a maximal quotient of C33⋊4C8 C32⋊3Dic9 He3⋊4Dic3 C34⋊C4 C62⋊Dic3
| action | f(x) | Disc(f) |
|---|---|---|
| 12T72 | x12+12x10-40x9+414x8-1416x7+3388x6-6552x5+8001x4-7448x3+7056x2+4704x+2744 | 287·316·714·172·13612·15427272 |
Matrix representation of C33⋊C4 ►in GL4(𝔽7) generated by
| 6 | 2 | 1 | 1 |
| 2 | 6 | 6 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 2 |
| 6 | 2 | 5 | 1 |
| 0 | 4 | 0 | 5 |
| 4 | 4 | 0 | 6 |
| 0 | 0 | 0 | 2 |
| 3 | 1 | 4 | 5 |
| 1 | 3 | 3 | 5 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 3 | 1 | 4 |
| 3 | 6 | 4 | 5 |
| 1 | 6 | 3 | 2 |
| 5 | 5 | 1 | 4 |
G:=sub<GL(4,GF(7))| [6,2,0,0,2,6,0,0,1,6,1,0,1,1,0,2],[6,0,4,0,2,4,4,0,5,0,0,0,1,5,6,2],[3,1,0,0,1,3,0,0,4,3,4,0,5,5,0,2],[1,3,1,5,3,6,6,5,1,4,3,1,4,5,2,4] >;
C33⋊C4 in GAP, Magma, Sage, TeX
C_3^3\rtimes C_4
% in TeX
G:=Group("C3^3:C4"); // GroupNames label
G:=SmallGroup(108,37);
// by ID
G=gap.SmallGroup(108,37);
# by ID
G:=PCGroup([5,-2,-2,-3,3,-3,10,302,67,323,248,1804]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1,b*c=c*b,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of C33⋊C4 in TeX
Character table of C33⋊C4 in TeX