metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊4S3, C23.30D6, (C2×C6)⋊8D4, C3⋊3C22≀C2, (C23×C6)⋊3C2, C6.63(C2×D4), C22⋊4(C3⋊D4), (C2×C6).61C23, C6.D4⋊13C2, (C22×S3)⋊2C22, (C2×Dic3)⋊3C22, C22.66(C22×S3), (C22×C6).42C22, (C2×C3⋊D4)⋊8C2, C2.26(C2×C3⋊D4), SmallGroup(96,160)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊4S3
G = < a,b,c,d,e,f | a2=b2=c2=d2=e3=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
Subgroups: 290 in 130 conjugacy classes, 41 normal (8 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, Dic3, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C2×D4, C24, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C22×C6, C22≀C2, C6.D4, C2×C3⋊D4, C23×C6, C24⋊4S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, C2×C3⋊D4, C24⋊4S3
Character table of C24⋊4S3
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 6O | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 12 | 2 | 12 | 12 | 12 | 2 | 2 | 2 | 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 | 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 | -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 | -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 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 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 | 1 | 1 | 1 | linear of order 2 |
| ρ6 | 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 | -1 | -1 | 1 | linear of order 2 |
| ρ7 | 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 | -1 | -1 | 1 | linear of order 2 |
| ρ8 | 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 | 1 | 1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 2 | orthogonal lifted from D4 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 2 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | -1 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | -1 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 2 | -2 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | √-3 | -√-3 | 1 | -√-3 | √-3 | -√-3 | 1 | 1 | √-3 | -1 | -1 | 1 | √-3 | -√-3 | -1 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | √-3 | -√-3 | √-3 | -√-3 | -√-3 | √-3 | -1 | √-3 | -√-3 | 1 | complex lifted from C3⋊D4 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | √-3 | -√-3 | 1 | -√-3 | √-3 | √-3 | -1 | -1 | -√-3 | 1 | 1 | 1 | -√-3 | √-3 | -1 | complex lifted from C3⋊D4 |
| ρ22 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | √-3 | -√-3 | -1 | √-3 | -√-3 | -1 | √-3 | -√-3 | -1 | -√-3 | √-3 | 1 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ23 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | √-3 | √-3 | -√-3 | -√-3 | √-3 | -√-3 | -1 | √-3 | -√-3 | 1 | complex lifted from C3⋊D4 |
| ρ24 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | -√-3 | √-3 | -1 | -√-3 | √-3 | -1 | -√-3 | √-3 | -1 | √-3 | -√-3 | 1 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ25 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | √-3 | -√-3 | -1 | √-3 | -√-3 | 1 | -√-3 | √-3 | 1 | √-3 | -√-3 | 1 | -1 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ26 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | -√-3 | -√-3 | √-3 | √-3 | -√-3 | √-3 | -1 | -√-3 | √-3 | 1 | complex lifted from C3⋊D4 |
| ρ27 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | -√-3 | √-3 | -1 | -√-3 | √-3 | 1 | √-3 | -√-3 | 1 | -√-3 | √-3 | 1 | -1 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ28 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | -√-3 | √-3 | 1 | √-3 | -√-3 | -√-3 | -1 | -1 | √-3 | 1 | 1 | 1 | √-3 | -√-3 | -1 | complex lifted from C3⋊D4 |
| ρ29 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | -√-3 | √-3 | -√-3 | √-3 | √-3 | -√-3 | -1 | -√-3 | √-3 | 1 | complex lifted from C3⋊D4 |
| ρ30 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -1 | 0 | 0 | 0 | -√-3 | √-3 | 1 | √-3 | -√-3 | √-3 | 1 | 1 | -√-3 | -1 | -1 | 1 | -√-3 | √-3 | -1 | complex lifted from C3⋊D4 |
►On 24 points - transitive group 24T116
(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 14)(2 13)(3 15)(4 17)(5 16)(6 18)(7 20)(8 19)(9 21)(10 23)(11 22)(12 24)
G:=sub<Sym(24)| (13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,14)(2,13)(3,15)(4,17)(5,16)(6,18)(7,20)(8,19)(9,21)(10,23)(11,22)(12,24)>;
G:=Group( (13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,14)(2,13)(3,15)(4,17)(5,16)(6,18)(7,20)(8,19)(9,21)(10,23)(11,22)(12,24) );
G=PermutationGroup([[(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,22),(14,23),(15,24),(16,19),(17,20),(18,21)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,14),(2,13),(3,15),(4,17),(5,16),(6,18),(7,20),(8,19),(9,21),(10,23),(11,22),(12,24)]])
G:=TransitiveGroup(24,116);
C24⋊4S3 is a maximal subgroup of
C24⋊6D6 C24.38D6 C24.41D6 C24.42D6 C24.67D6 S3×C22≀C2 C24⋊7D6 C24.45D6 C24.46D6 C24⋊9D6 C24.47D6 C24.83D6 D4×C3⋊D4 C24⋊12D6 C24.52D6 C24.53D6 C24⋊4D9 C23.D18 (C22×S3)⋊A4 C62⋊4D4 C62⋊5D4 C62⋊24D4 C24⋊D9 (C2×C6)⋊4S4 (C2×C6)⋊S4 (C2×C30)⋊D4 (C2×C6)⋊8D20 C24⋊5D15 C24⋊D15
C24⋊4S3 is a maximal quotient of
C24.73D6 C24.76D6 (C2×C6)⋊8D8 (C3×D4).31D4 C24.31D6 C24.32D6 (C3×Q8)⋊13D4 (C2×C6)⋊8Q16 C22.52(S3×Q8) (C22×Q8)⋊9S3 (C3×D4)⋊14D4 (C3×D4).32D4 2+ 1+4⋊6S3 2+ 1+4.4S3 2+ 1+4.5S3 2+ 1+4⋊7S3 2- 1+4⋊4S3 2- 1+4.2S3 C25.4S3 C24⋊4D9 C62⋊4D4 C62⋊5D4 C62⋊24D4 (C2×C30)⋊D4 (C2×C6)⋊8D20 C24⋊5D15
Matrix representation of C24⋊4S3 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 9 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[9,0,0,0,0,3,0,0,0,0,3,0,0,0,0,9],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C24⋊4S3 in GAP, Magma, Sage, TeX
C_2^4\rtimes_4S_3
% in TeX
G:=Group("C2^4:4S3"); // GroupNames label
G:=SmallGroup(96,160);
// by ID
G=gap.SmallGroup(96,160);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,218,2309]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^3=f^2=1,a*b=b*a,a*c=c*a,f*a*f=a*d=d*a,a*e=e*a,f*b*f=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations
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