metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊3S3, (C4×C12)⋊1C2, D6⋊C4.1C2, (C2×C4).63D6, Dic3⋊C4⋊1C2, C6.6(C4○D4), C3⋊1(C42⋊2C2), C2.8(C4○D12), (C2×C6).17C23, (C2×C12).75C22, (C22×S3).3C22, C22.38(C22×S3), (C2×Dic3).4C22, SmallGroup(96,83)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊3S3
G = < a,b,c,d | a4=b4=c3=d2=1, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b-1, dcd=c-1 >
Subgroups: 138 in 60 conjugacy classes, 29 normal (8 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×Dic3, C2×C12, C22×S3, C42⋊2C2, Dic3⋊C4, D6⋊C4, C4×C12, C42⋊3S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, C42⋊2C2, C4○D12, C42⋊3S3
Character table of C42⋊3S3
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | -1 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 0 | -1 | -2 | -2 | -2 | 2 | 2 | -2 | 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 | 2 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | -2i | 0 | 0 | 2i | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2i | 0 | 0 | -2i | 0 | 0 | 2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | -2i | 0 | 0 | -2i | 0 | 0 | 0 | 2i | 2i | 0 | complex lifted from C4○D4 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 2i | 0 | 0 | -2i | complex lifted from C4○D4 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2i | 0 | 0 | 2i | 0 | 0 | -2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2i | 0 | 0 | 2i | 0 | 0 | 0 | -2i | -2i | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√3 | i | -√-3 | √3 | -i | √-3 | √3 | -√3 | -i | -√-3 | √-3 | i | complex lifted from C4○D12 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √3 | -i | -√-3 | -√3 | i | √-3 | -√3 | √3 | i | -√-3 | √-3 | -i | complex lifted from C4○D12 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -i | √-3 | √3 | -i | -√-3 | -√3 | i | i | √-3 | -√3 | √3 | -√-3 | complex lifted from C4○D12 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √3 | i | √-3 | -√3 | -i | -√-3 | -√3 | √3 | -i | √-3 | -√-3 | i | complex lifted from C4○D12 |
| ρ23 | 2 | 2 | -2 | -2 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√3 | -i | √-3 | √3 | i | -√-3 | √3 | -√3 | i | √-3 | -√-3 | -i | complex lifted from C4○D12 |
| ρ24 | 2 | -2 | -2 | 2 | 0 | -1 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | -1 | 1 | 1 | √-3 | √3 | i | -√-3 | √3 | i | √-3 | -√-3 | -√3 | -i | -i | -√3 | complex lifted from C4○D12 |
| ρ25 | 2 | -2 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -i | -√-3 | -√3 | -i | √-3 | √3 | i | i | -√-3 | √3 | -√3 | √-3 | complex lifted from C4○D12 |
| ρ26 | 2 | -2 | -2 | 2 | 0 | -1 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | -1 | 1 | 1 | -√-3 | -√3 | i | √-3 | -√3 | i | -√-3 | √-3 | √3 | -i | -i | √3 | complex lifted from C4○D12 |
| ρ27 | 2 | -2 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | i | -√-3 | √3 | i | √-3 | -√3 | -i | -i | -√-3 | -√3 | √3 | √-3 | complex lifted from C4○D12 |
| ρ28 | 2 | -2 | -2 | 2 | 0 | -1 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | -1 | 1 | 1 | -√-3 | √3 | -i | √-3 | √3 | -i | -√-3 | √-3 | -√3 | i | i | -√3 | complex lifted from C4○D12 |
| ρ29 | 2 | -2 | 2 | -2 | 0 | -1 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | i | √-3 | -√3 | i | -√-3 | √3 | -i | -i | √-3 | √3 | -√3 | -√-3 | complex lifted from C4○D12 |
| ρ30 | 2 | -2 | -2 | 2 | 0 | -1 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | -1 | 1 | 1 | √-3 | -√3 | -i | -√-3 | -√3 | -i | √-3 | -√-3 | √3 | i | i | √3 | complex lifted from C4○D12 |
►On 48 points
(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 26 43 17)(2 27 44 18)(3 28 41 19)(4 25 42 20)(5 22 34 16)(6 23 35 13)(7 24 36 14)(8 21 33 15)(9 30 47 39)(10 31 48 40)(11 32 45 37)(12 29 46 38)
(1 23 9)(2 24 10)(3 21 11)(4 22 12)(5 38 20)(6 39 17)(7 40 18)(8 37 19)(13 47 43)(14 48 44)(15 45 41)(16 46 42)(25 34 29)(26 35 30)(27 36 31)(28 33 32)
(2 44)(4 42)(5 40)(6 32)(7 38)(8 30)(9 23)(10 14)(11 21)(12 16)(13 47)(15 45)(17 28)(18 20)(19 26)(22 46)(24 48)(25 27)(29 36)(31 34)(33 39)(35 37)
G:=sub<Sym(48)| (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), (1,26,43,17)(2,27,44,18)(3,28,41,19)(4,25,42,20)(5,22,34,16)(6,23,35,13)(7,24,36,14)(8,21,33,15)(9,30,47,39)(10,31,48,40)(11,32,45,37)(12,29,46,38), (1,23,9)(2,24,10)(3,21,11)(4,22,12)(5,38,20)(6,39,17)(7,40,18)(8,37,19)(13,47,43)(14,48,44)(15,45,41)(16,46,42)(25,34,29)(26,35,30)(27,36,31)(28,33,32), (2,44)(4,42)(5,40)(6,32)(7,38)(8,30)(9,23)(10,14)(11,21)(12,16)(13,47)(15,45)(17,28)(18,20)(19,26)(22,46)(24,48)(25,27)(29,36)(31,34)(33,39)(35,37)>;
G:=Group( (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), (1,26,43,17)(2,27,44,18)(3,28,41,19)(4,25,42,20)(5,22,34,16)(6,23,35,13)(7,24,36,14)(8,21,33,15)(9,30,47,39)(10,31,48,40)(11,32,45,37)(12,29,46,38), (1,23,9)(2,24,10)(3,21,11)(4,22,12)(5,38,20)(6,39,17)(7,40,18)(8,37,19)(13,47,43)(14,48,44)(15,45,41)(16,46,42)(25,34,29)(26,35,30)(27,36,31)(28,33,32), (2,44)(4,42)(5,40)(6,32)(7,38)(8,30)(9,23)(10,14)(11,21)(12,16)(13,47)(15,45)(17,28)(18,20)(19,26)(22,46)(24,48)(25,27)(29,36)(31,34)(33,39)(35,37) );
G=PermutationGroup([[(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)], [(1,26,43,17),(2,27,44,18),(3,28,41,19),(4,25,42,20),(5,22,34,16),(6,23,35,13),(7,24,36,14),(8,21,33,15),(9,30,47,39),(10,31,48,40),(11,32,45,37),(12,29,46,38)], [(1,23,9),(2,24,10),(3,21,11),(4,22,12),(5,38,20),(6,39,17),(7,40,18),(8,37,19),(13,47,43),(14,48,44),(15,45,41),(16,46,42),(25,34,29),(26,35,30),(27,36,31),(28,33,32)], [(2,44),(4,42),(5,40),(6,32),(7,38),(8,30),(9,23),(10,14),(11,21),(12,16),(13,47),(15,45),(17,28),(18,20),(19,26),(22,46),(24,48),(25,27),(29,36),(31,34),(33,39),(35,37)]])
C42⋊3S3 is a maximal subgroup of
C42.277D6 C42⋊12D6 C42.95D6 C42.96D6 C42.98D6 C42.104D6 C42⋊18D6 C42⋊19D6 C42.118D6 C42.122D6 C42.132D6 C42.133D6 C42.134D6 C42.137D6 C42⋊22D6 C42.150D6 C42.154D6 S3×C42⋊2C2 C42.189D6 C42⋊27D6 C42.165D6 C42⋊30D6 C42.180D6 C42⋊3D9 C42⋊C3⋊S3 C62.29C23 C62.38C23 C122⋊2C2 C5⋊(C42⋊3S3) (C4×Dic5)⋊S3 C42⋊3D15
C42⋊3S3 is a maximal quotient of
(C2×Dic3).9D4 (C22×C4).30D6 C6.(C4⋊D4) (C22×C4).37D6 (C2×C42).6S3 C42⋊7Dic3 (C2×C42)⋊3S3 C42⋊3D9 C62.29C23 C62.38C23 C122⋊2C2 C5⋊(C42⋊3S3) (C4×Dic5)⋊S3 C42⋊3D15
Matrix representation of C42⋊3S3 ►in GL4(𝔽13) generated by
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 10 | 6 |
| 0 | 0 | 7 | 3 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [5,0,0,0,0,5,0,0,0,0,10,7,0,0,6,3],[0,1,0,0,1,0,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,12],[1,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0] >;
C42⋊3S3 in GAP, Magma, Sage, TeX
C_4^2\rtimes_3S_3
% in TeX
G:=Group("C4^2:3S3"); // GroupNames label
G:=SmallGroup(96,83);
// by ID
G=gap.SmallGroup(96,83);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,55,506,86,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^3=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations
Export