Aliases: C42⋊He3, C122⋊1C3, C62.3A4, C32⋊(C42⋊C3), (C4×C12).5C32, C22.(C32⋊A4), (C3×C42⋊C3)⋊C3, C3.5(C3×C42⋊C3), (C2×C6).10(C3×A4), SmallGroup(432,103)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊He3
G = < a,b,c,d,e | a4=b4=c3=d3=e3=1, ab=ba, ac=ca, ad=da, eae-1=ab-1, bc=cb, bd=db, ebe-1=a-1b2, cd=dc, ece-1=cd-1, de=ed >
Subgroups: 363 in 57 conjugacy classes, 13 normal (10 characteristic)
C1, C2, C3, C3, C4, C22, C6, C2×C4, C32, C32, C12, A4, C2×C6, C2×C6, C42, C3×C6, C2×C12, He3, C3×C12, C3×A4, C62, C42⋊C3, C4×C12, C4×C12, C6×C12, C32⋊A4, C3×C42⋊C3, C122, C42⋊He3
Quotients: C1, C3, C32, A4, He3, C3×A4, C42⋊C3, C32⋊A4, C3×C42⋊C3, C42⋊He3
►On 36 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)
(1 4 2 3)(5 8 6 7)(9 12 10 11)(13 16 15 14)(17 20 19 18)(21 24 23 22)
(13 24 19)(14 21 20)(15 22 17)(16 23 18)(25 29 33)(26 30 34)(27 31 35)(28 32 36)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 24 19)(14 21 20)(15 22 17)(16 23 18)(25 33 29)(26 34 30)(27 35 31)(28 36 32)
(1 27 13)(2 25 15)(3 26 16)(4 28 14)(5 31 19)(6 29 17)(7 30 18)(8 32 20)(9 35 24)(10 33 22)(11 34 23)(12 36 21)
G:=sub<Sym(36)| (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), (1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,15,14)(17,20,19,18)(21,24,23,22), (13,24,19)(14,21,20)(15,22,17)(16,23,18)(25,29,33)(26,30,34)(27,31,35)(28,32,36), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,24,19)(14,21,20)(15,22,17)(16,23,18)(25,33,29)(26,34,30)(27,35,31)(28,36,32), (1,27,13)(2,25,15)(3,26,16)(4,28,14)(5,31,19)(6,29,17)(7,30,18)(8,32,20)(9,35,24)(10,33,22)(11,34,23)(12,36,21)>;
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), (1,4,2,3)(5,8,6,7)(9,12,10,11)(13,16,15,14)(17,20,19,18)(21,24,23,22), (13,24,19)(14,21,20)(15,22,17)(16,23,18)(25,29,33)(26,30,34)(27,31,35)(28,32,36), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,24,19)(14,21,20)(15,22,17)(16,23,18)(25,33,29)(26,34,30)(27,35,31)(28,36,32), (1,27,13)(2,25,15)(3,26,16)(4,28,14)(5,31,19)(6,29,17)(7,30,18)(8,32,20)(9,35,24)(10,33,22)(11,34,23)(12,36,21) );
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)], [(1,4,2,3),(5,8,6,7),(9,12,10,11),(13,16,15,14),(17,20,19,18),(21,24,23,22)], [(13,24,19),(14,21,20),(15,22,17),(16,23,18),(25,29,33),(26,30,34),(27,31,35),(28,32,36)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,24,19),(14,21,20),(15,22,17),(16,23,18),(25,33,29),(26,34,30),(27,35,31),(28,36,32)], [(1,27,13),(2,25,15),(3,26,16),(4,28,14),(5,31,19),(6,29,17),(7,30,18),(8,32,20),(9,35,24),(10,33,22),(11,34,23),(12,36,21)]])
56 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 4A | 4B | 4C | 4D | 6A | ··· | 6H | 12A | ··· | 12AF |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 48 | ··· | 48 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
56 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C3 | C3 | A4 | He3 | C3×A4 | C42⋊C3 | C32⋊A4 | C3×C42⋊C3 | C42⋊He3 |
| kernel | C42⋊He3 | C3×C42⋊C3 | C122 | C62 | C42 | C2×C6 | C32 | C22 | C3 | C1 |
| # reps | 1 | 6 | 2 | 1 | 2 | 2 | 4 | 6 | 8 | 24 |
Matrix representation of C42⋊He3 ►in GL3(𝔽13) generated by
| 12 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 8 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 3 |
| 9 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(13))| [12,0,0,0,8,0,0,0,8],[8,0,0,0,5,0,0,0,1],[1,0,0,0,9,0,0,0,3],[9,0,0,0,9,0,0,0,9],[0,0,1,1,0,0,0,1,0] >;
C42⋊He3 in GAP, Magma, Sage, TeX
C_4^2\rtimes {\rm He}_3 % in TeX
G:=Group("C4^2:He3"); // GroupNames label
G:=SmallGroup(432,103);
// by ID
G=gap.SmallGroup(432,103);
# by ID
G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,169,1515,360,10399,102,9077,15882]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^3=d^3=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^-1,b*c=c*b,b*d=d*b,e*b*e^-1=a^-1*b^2,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations