Aliases: (C6xC12).1C4, C3:Dic3.9D4, C62.8(C2xC4), C2.5(C62:C4), C32:2(C4.10D4), C62.C4.2C2, (C2xC4).(C32:C4), (C2xC3:Dic3).3C4, C22.3(C2xC32:C4), (C2xC32:4Q8).2C2, (C3xC6).14(C22:C4), (C2xC3: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, C2xC4, C2xC4, Q8, C32, Dic3, C12, C2xC6, M4(2), C2xQ8, C3xC6, C3xC6, Dic6, C2xDic3, C2xC12, C4.10D4, C3:Dic3, C3:Dic3, C3xC12, C62, C2xDic6, C32:2C8, C32:4Q8, C2xC3:Dic3, C6xC12, C62.C4, C2xC32:4Q8, C3:Dic3.D4
Quotients: C1, C2, C4, C22, C2xC4, D4, C22:C4, C4.10D4, C32:C4, C2xC32: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 C2xC32: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 C2xC32: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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