non-abelian, soluble, monomial
Aliases: C4.12S3≀C2, (C3×C12).4D4, C3⋊S3.2Q16, C32⋊2Q8⋊1C4, C3⋊S3.3SD16, C32⋊2(Q8⋊C4), Dic3.D6.2C2, C12.29D6.2C2, C2.6(S32⋊C4), (C2×C3⋊S3).6D4, C4⋊(C32⋊C4).1C2, (C4×C3⋊S3).4C22, C3⋊Dic3.6(C2×C4), (C3×C6).5(C22⋊C4), SmallGroup(288,378)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊S3.2Q16
G = < a,b,c,d,e | a3=b3=c2=d8=1, e2=d4, ab=ba, cac=dbd-1=a-1, dad-1=cbc=ebe-1=b-1, ae=ea, cd=dc, ce=ec, ede-1=cd-1 >
Subgroups: 368 in 72 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2×C4, Q8, C32, Dic3, C12, D6, C4⋊C4, C2×C8, C2×Q8, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, C4×S3, C3×Q8, Q8⋊C4, C3×Dic3, C3⋊Dic3, C3×C12, C32⋊C4, C2×C3⋊S3, S3×C8, S3×Q8, C3×C3⋊C8, C6.D6, C32⋊2Q8, C3×Dic6, C4×C3⋊S3, C2×C32⋊C4, C12.29D6, C4⋊(C32⋊C4), Dic3.D6, C3⋊S3.2Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, SD16, Q16, Q8⋊C4, S3≀C2, S32⋊C4, C3⋊S3.2Q16
Character table of C3⋊S3.2Q16
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 9 | 9 | 4 | 4 | 2 | 12 | 12 | 18 | 36 | 36 | 4 | 4 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 24 | 24 | 12 | 12 | 12 | 12 | |
| 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 | -i | i | 1 | 1 | -i | -i | i | i | -1 | -1 | -1 | 1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -i | i | 1 | 1 | i | i | -i | -i | -1 | -1 | -1 | -1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | i | -i | 1 | 1 | i | i | -i | -i | -1 | -1 | -1 | 1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | i | -i | 1 | 1 | -i | -i | i | i | -1 | -1 | -1 | -1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | √2 | -√2 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ12 | 2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -√2 | √2 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ13 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -√-2 | √-2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | complex lifted from SD16 |
| ρ14 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | √-2 | -√-2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | complex lifted from SD16 |
| ρ15 | 4 | 4 | 0 | 0 | -2 | 1 | 4 | -2 | -2 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ16 | 4 | 4 | 0 | 0 | -2 | 1 | 4 | 2 | 2 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ17 | 4 | 4 | 0 | 0 | -2 | 1 | -4 | 2 | -2 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S32⋊C4 |
| ρ18 | 4 | 4 | 0 | 0 | 1 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3≀C2 |
| ρ19 | 4 | 4 | 0 | 0 | 1 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | -2 | -2 | -2 | -2 | 1 | 1 | -2 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S3≀C2 |
| ρ20 | 4 | 4 | 0 | 0 | -2 | 1 | -4 | -2 | 2 | 0 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S32⋊C4 |
| ρ21 | 4 | 4 | 0 | 0 | 1 | -2 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | 2i | 2i | -2i | -2i | -1 | -1 | 2 | 0 | 0 | -i | -i | i | i | complex lifted from S32⋊C4 |
| ρ22 | 4 | 4 | 0 | 0 | 1 | -2 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | -2i | -2i | 2i | 2i | -1 | -1 | 2 | 0 | 0 | i | i | -i | -i | complex lifted from S32⋊C4 |
| ρ23 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | 2ζ83 | 2ζ87 | 2ζ85 | 2ζ8 | -3i | 3i | 0 | 0 | 0 | ζ83 | ζ87 | ζ85 | ζ8 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | 2ζ8 | 2ζ85 | 2ζ87 | 2ζ83 | 3i | -3i | 0 | 0 | 0 | ζ8 | ζ85 | ζ87 | ζ83 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | 2ζ87 | 2ζ83 | 2ζ8 | 2ζ85 | -3i | 3i | 0 | 0 | 0 | ζ87 | ζ83 | ζ8 | ζ85 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | 2ζ85 | 2ζ8 | 2ζ83 | 2ζ87 | 3i | -3i | 0 | 0 | 0 | ζ85 | ζ8 | ζ83 | ζ87 | complex faithful |
| ρ27 | 8 | -8 | 0 | 0 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(1 28 38)(2 29 39)(3 30 40)(4 31 33)(5 32 34)(6 25 35)(7 26 36)(8 27 37)(9 46 19)(10 20 47)(11 48 21)(12 22 41)(13 42 23)(14 24 43)(15 44 17)(16 18 45)
(1 38 28)(2 39 29)(3 40 30)(4 33 31)(5 34 32)(6 35 25)(7 36 26)(8 37 27)(9 46 19)(10 20 47)(11 48 21)(12 22 41)(13 42 23)(14 24 43)(15 44 17)(16 18 45)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)
(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 13 5 9)(2 16 6 12)(3 11 7 15)(4 14 8 10)(17 40 21 36)(18 25 22 29)(19 38 23 34)(20 31 24 27)(26 44 30 48)(28 42 32 46)(33 43 37 47)(35 41 39 45)
G:=sub<Sym(48)| (1,28,38)(2,29,39)(3,30,40)(4,31,33)(5,32,34)(6,25,35)(7,26,36)(8,27,37)(9,46,19)(10,20,47)(11,48,21)(12,22,41)(13,42,23)(14,24,43)(15,44,17)(16,18,45), (1,38,28)(2,39,29)(3,40,30)(4,33,31)(5,34,32)(6,35,25)(7,36,26)(8,37,27)(9,46,19)(10,20,47)(11,48,21)(12,22,41)(13,42,23)(14,24,43)(15,44,17)(16,18,45), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38), (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), (1,13,5,9)(2,16,6,12)(3,11,7,15)(4,14,8,10)(17,40,21,36)(18,25,22,29)(19,38,23,34)(20,31,24,27)(26,44,30,48)(28,42,32,46)(33,43,37,47)(35,41,39,45)>;
G:=Group( (1,28,38)(2,29,39)(3,30,40)(4,31,33)(5,32,34)(6,25,35)(7,26,36)(8,27,37)(9,46,19)(10,20,47)(11,48,21)(12,22,41)(13,42,23)(14,24,43)(15,44,17)(16,18,45), (1,38,28)(2,39,29)(3,40,30)(4,33,31)(5,34,32)(6,35,25)(7,36,26)(8,37,27)(9,46,19)(10,20,47)(11,48,21)(12,22,41)(13,42,23)(14,24,43)(15,44,17)(16,18,45), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38), (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), (1,13,5,9)(2,16,6,12)(3,11,7,15)(4,14,8,10)(17,40,21,36)(18,25,22,29)(19,38,23,34)(20,31,24,27)(26,44,30,48)(28,42,32,46)(33,43,37,47)(35,41,39,45) );
G=PermutationGroup([[(1,28,38),(2,29,39),(3,30,40),(4,31,33),(5,32,34),(6,25,35),(7,26,36),(8,27,37),(9,46,19),(10,20,47),(11,48,21),(12,22,41),(13,42,23),(14,24,43),(15,44,17),(16,18,45)], [(1,38,28),(2,39,29),(3,40,30),(4,33,31),(5,34,32),(6,35,25),(7,36,26),(8,37,27),(9,46,19),(10,20,47),(11,48,21),(12,22,41),(13,42,23),(14,24,43),(15,44,17),(16,18,45)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,48),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,39),(26,40),(27,33),(28,34),(29,35),(30,36),(31,37),(32,38)], [(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)], [(1,13,5,9),(2,16,6,12),(3,11,7,15),(4,14,8,10),(17,40,21,36),(18,25,22,29),(19,38,23,34),(20,31,24,27),(26,44,30,48),(28,42,32,46),(33,43,37,47),(35,41,39,45)]])
Matrix representation of C3⋊S3.2Q16 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 57 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 61 | 0 | 0 | 0 | 0 |
| 6 | 61 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,57,0,0,0,0,32,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,6,0,0,0,0,61,61,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C3⋊S3.2Q16 in GAP, Magma, Sage, TeX
C_3\rtimes S_3._2Q_{16} % in TeX
G:=Group("C3:S3.2Q16"); // GroupNames label
G:=SmallGroup(288,378);
// by ID
G=gap.SmallGroup(288,378);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,85,64,422,100,675,80,2693,2028,691,797,2372]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=1,e^2=d^4,a*b=b*a,c*a*c=d*b*d^-1=a^-1,d*a*d^-1=c*b*c=e*b*e^-1=b^-1,a*e=e*a,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations
Export