Aliases: (C6×C12).1C4, C3⋊Dic3.9D4, C62.8(C2×C4), C2.5(C62⋊C4), C32⋊2(C4.10D4), C62.C4.2C2, (C2×C4).(C32⋊C4), (C2×C3⋊Dic3).3C4, C22.3(C2×C32⋊C4), (C2×C32⋊4Q8).2C2, (C3×C6).14(C22⋊C4), (C2×C3⋊Dic3).7C22, SmallGroup(288,428)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊Dic3.D4
G = < a,b,c,d,e | a3=b6=1, c2=d4=b3, e2=dcd-1=b3c, ab=ba, cac-1=a-1, dad-1=eae-1=ab4, cbc-1=b-1, dbd-1=ebe-1=ab-1, ce=ec, ede-1=b3cd3 >
Subgroups: 360 in 76 conjugacy classes, 16 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C6, C8, C2×C4, C2×C4, Q8, C32, Dic3, C12, C2×C6, M4(2), C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C4.10D4, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C2×Dic6, C32⋊2C8, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C62.C4, C2×C32⋊4Q8, C3⋊Dic3.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, C4.10D4, C32⋊C4, C2×C32⋊C4, C62⋊C4, C3⋊Dic3.D4
Character table of C3⋊Dic3.D4
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 18 | 18 | 36 | 4 | 4 | 4 | 4 | 4 | 4 | 36 | 36 | 36 | 36 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | i | i | -i | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | i | -i | -i | i | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | -2 | 2 | 2 | 0 | -2 | 2 | 0 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 2 | 2 | 0 | 2 | -2 | 0 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | -4 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 3 | 3 | 0 | 0 | -3 | orthogonal lifted from C62⋊C4 |
| ρ12 | 4 | 4 | -4 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | -3 | -3 | 0 | 0 | 3 | orthogonal lifted from C62⋊C4 |
| ρ13 | 4 | 4 | -4 | -2 | 1 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 3 | 0 | -3 | 0 | 0 | 3 | -3 | 0 | orthogonal lifted from C62⋊C4 |
| ρ14 | 4 | 4 | 4 | 1 | -2 | 4 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 1 | -2 | 1 | 1 | -2 | -2 | 1 | orthogonal lifted from C32⋊C4 |
| ρ15 | 4 | 4 | 4 | -2 | 1 | -4 | 0 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | -1 | -1 | 2 | orthogonal lifted from C2×C32⋊C4 |
| ρ16 | 4 | 4 | 4 | -2 | 1 | 4 | 0 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | -2 | -2 | 1 | 1 | -2 | orthogonal lifted from C32⋊C4 |
| ρ17 | 4 | 4 | -4 | -2 | 1 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | -3 | 0 | 3 | 0 | 0 | -3 | 3 | 0 | orthogonal lifted from C62⋊C4 |
| ρ18 | 4 | 4 | 4 | 1 | -2 | -4 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | orthogonal lifted from C2×C32⋊C4 |
| ρ19 | 4 | -4 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.10D4, Schur index 2 |
| ρ20 | 4 | -4 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 3 | -1 | -3 | 0 | 0 | 0 | 0 | √3 | 0 | -√3 | 2√3 | -2√3 | -√3 | √3 | 0 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -3 | -1 | 3 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -√3 | -2√3 | -√3 | √3 | 0 | 2√3 | √3 | symplectic faithful, Schur index 2 |
| ρ22 | 4 | -4 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -3 | -1 | 3 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | √3 | 2√3 | √3 | -√3 | 0 | -2√3 | -√3 | symplectic faithful, Schur index 2 |
| ρ23 | 4 | -4 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 3 | -1 | -3 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2√3 | -√3 | 0 | √3 | -√3 | 2√3 | 0 | √3 | symplectic faithful, Schur index 2 |
| ρ24 | 4 | -4 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 3 | -1 | -3 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2√3 | √3 | 0 | -√3 | √3 | -2√3 | 0 | -√3 | symplectic faithful, Schur index 2 |
| ρ25 | 4 | -4 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -3 | -1 | 3 | 0 | 0 | 0 | 0 | √3 | -2√3 | √3 | 0 | 0 | -√3 | -√3 | 2√3 | symplectic faithful, Schur index 2 |
| ρ26 | 4 | -4 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -3 | -1 | 3 | 0 | 0 | 0 | 0 | -√3 | 2√3 | -√3 | 0 | 0 | √3 | √3 | -2√3 | symplectic faithful, Schur index 2 |
| ρ27 | 4 | -4 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 3 | -1 | -3 | 0 | 0 | 0 | 0 | -√3 | 0 | √3 | -2√3 | 2√3 | √3 | -√3 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(1 47 13)(3 15 41)(5 43 9)(7 11 45)(18 32 33)(20 35 26)(22 28 37)(24 39 30)
(1 9 47 5 13 43)(2 10 48 6 14 44)(3 45 15 7 41 11)(4 46 16 8 42 12)(17 27 40 21 31 36)(18 37 32 22 33 28)(19 38 25 23 34 29)(20 30 35 24 26 39)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(25 31 29 27)(26 28 30 32)(33 35 37 39)(34 40 38 36)(41 43 45 47)(42 48 46 44)
(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 29 7 31 5 25 3 27)(2 30 4 28 6 26 8 32)(9 19 15 21 13 23 11 17)(10 20 12 18 14 24 16 22)(33 48 39 42 37 44 35 46)(34 41 36 47 38 45 40 43)
G:=sub<Sym(48)| (1,47,13)(3,15,41)(5,43,9)(7,11,45)(18,32,33)(20,35,26)(22,28,37)(24,39,30), (1,9,47,5,13,43)(2,10,48,6,14,44)(3,45,15,7,41,11)(4,46,16,8,42,12)(17,27,40,21,31,36)(18,37,32,22,33,28)(19,38,25,23,34,29)(20,30,35,24,26,39), (1,3,5,7)(2,8,6,4)(9,11,13,15)(10,16,14,12)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32)(33,35,37,39)(34,40,38,36)(41,43,45,47)(42,48,46,44), (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), (1,29,7,31,5,25,3,27)(2,30,4,28,6,26,8,32)(9,19,15,21,13,23,11,17)(10,20,12,18,14,24,16,22)(33,48,39,42,37,44,35,46)(34,41,36,47,38,45,40,43)>;
G:=Group( (1,47,13)(3,15,41)(5,43,9)(7,11,45)(18,32,33)(20,35,26)(22,28,37)(24,39,30), (1,9,47,5,13,43)(2,10,48,6,14,44)(3,45,15,7,41,11)(4,46,16,8,42,12)(17,27,40,21,31,36)(18,37,32,22,33,28)(19,38,25,23,34,29)(20,30,35,24,26,39), (1,3,5,7)(2,8,6,4)(9,11,13,15)(10,16,14,12)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32)(33,35,37,39)(34,40,38,36)(41,43,45,47)(42,48,46,44), (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), (1,29,7,31,5,25,3,27)(2,30,4,28,6,26,8,32)(9,19,15,21,13,23,11,17)(10,20,12,18,14,24,16,22)(33,48,39,42,37,44,35,46)(34,41,36,47,38,45,40,43) );
G=PermutationGroup([[(1,47,13),(3,15,41),(5,43,9),(7,11,45),(18,32,33),(20,35,26),(22,28,37),(24,39,30)], [(1,9,47,5,13,43),(2,10,48,6,14,44),(3,45,15,7,41,11),(4,46,16,8,42,12),(17,27,40,21,31,36),(18,37,32,22,33,28),(19,38,25,23,34,29),(20,30,35,24,26,39)], [(1,3,5,7),(2,8,6,4),(9,11,13,15),(10,16,14,12),(17,23,21,19),(18,20,22,24),(25,31,29,27),(26,28,30,32),(33,35,37,39),(34,40,38,36),(41,43,45,47),(42,48,46,44)], [(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)], [(1,29,7,31,5,25,3,27),(2,30,4,28,6,26,8,32),(9,19,15,21,13,23,11,17),(10,20,12,18,14,24,16,22),(33,48,39,42,37,44,35,46),(34,41,36,47,38,45,40,43)]])
Matrix representation of C3⋊Dic3.D4 ►in GL4(𝔽73) generated by
| 72 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 72 | 1 | 0 | 0 |
| 0 | 0 | 1 | 72 |
| 0 | 0 | 1 | 0 |
| 61 | 63 | 0 | 0 |
| 51 | 12 | 0 | 0 |
| 0 | 0 | 12 | 10 |
| 0 | 0 | 22 | 61 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 61 | 63 | 0 | 0 |
| 51 | 12 | 0 | 0 |
| 0 | 0 | 66 | 14 |
| 0 | 0 | 59 | 7 |
| 68 | 19 | 0 | 0 |
| 14 | 5 | 0 | 0 |
G:=sub<GL(4,GF(73))| [72,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,1,1,0,0,0,0,1,1,0,0,72,0],[61,51,0,0,63,12,0,0,0,0,12,22,0,0,10,61],[0,0,61,51,0,0,63,12,1,0,0,0,0,1,0,0],[0,0,68,14,0,0,19,5,66,59,0,0,14,7,0,0] >;
C3⋊Dic3.D4 in GAP, Magma, Sage, TeX
C_3\rtimes {\rm Dic}_3.D_4 % in TeX
G:=Group("C3:Dic3.D4"); // GroupNames label
G:=SmallGroup(288,428);
// by ID
G=gap.SmallGroup(288,428);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,120,219,100,675,9413,691,12550,2372]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^6=1,c^2=d^4=b^3,e^2=d*c*d^-1=b^3*c,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=e*a*e^-1=a*b^4,c*b*c^-1=b^-1,d*b*d^-1=e*b*e^-1=a*b^-1,c*e=e*c,e*d*e^-1=b^3*c*d^3>;
// generators/relations
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