non-abelian, supersoluble, monomial
Aliases: He3⋊5D4, C32⋊4D12, (C3×C12)⋊2S3, C4⋊(He3⋊C2), (C4×He3)⋊2C2, (C3×C6).18D6, C12.8(C3⋊S3), C3.2(C12⋊S3), (C2×He3).13C22, C6.28(C2×C3⋊S3), (C2×He3⋊C2)⋊2C2, C2.4(C2×He3⋊C2), SmallGroup(216,68)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊5D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 358 in 88 conjugacy classes, 24 normal (10 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, D4, C32, C12, C12, D6, C2×C6, C3×S3, C3×C6, D12, C3×D4, He3, C3×C12, S3×C6, He3⋊C2, C2×He3, C3×D12, C4×He3, C2×He3⋊C2, He3⋊5D4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊S3, D12, C2×C3⋊S3, He3⋊C2, C12⋊S3, C2×He3⋊C2, He3⋊5D4
►On 36 points
(1 16 7)(2 13 8)(3 14 5)(4 15 6)(9 29 18)(10 30 19)(11 31 20)(12 32 17)(21 27 33)(22 28 34)(23 25 35)(24 26 36)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 11)(6 36 12)(7 33 9)(8 34 10)(13 28 19)(14 25 20)(15 26 17)(16 27 18)
(1 33 27)(2 34 28)(3 35 25)(4 36 26)(5 14 31)(6 15 32)(7 16 29)(8 13 30)(9 18 21)(10 19 22)(11 20 23)(12 17 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)(25 26 27 28)(29 30 31 32)(33 34 35 36)
(1 4)(2 3)(5 13)(6 16)(7 15)(8 14)(9 17)(10 20)(11 19)(12 18)(21 24)(22 23)(25 34)(26 33)(27 36)(28 35)(29 32)(30 31)
G:=sub<Sym(36)| (1,16,7)(2,13,8)(3,14,5)(4,15,6)(9,29,18)(10,30,19)(11,31,20)(12,32,17)(21,27,33)(22,28,34)(23,25,35)(24,26,36), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,11)(6,36,12)(7,33,9)(8,34,10)(13,28,19)(14,25,20)(15,26,17)(16,27,18), (1,33,27)(2,34,28)(3,35,25)(4,36,26)(5,14,31)(6,15,32)(7,16,29)(8,13,30)(9,18,21)(10,19,22)(11,20,23)(12,17,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,4)(2,3)(5,13)(6,16)(7,15)(8,14)(9,17)(10,20)(11,19)(12,18)(21,24)(22,23)(25,34)(26,33)(27,36)(28,35)(29,32)(30,31)>;
G:=Group( (1,16,7)(2,13,8)(3,14,5)(4,15,6)(9,29,18)(10,30,19)(11,31,20)(12,32,17)(21,27,33)(22,28,34)(23,25,35)(24,26,36), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,11)(6,36,12)(7,33,9)(8,34,10)(13,28,19)(14,25,20)(15,26,17)(16,27,18), (1,33,27)(2,34,28)(3,35,25)(4,36,26)(5,14,31)(6,15,32)(7,16,29)(8,13,30)(9,18,21)(10,19,22)(11,20,23)(12,17,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,4)(2,3)(5,13)(6,16)(7,15)(8,14)(9,17)(10,20)(11,19)(12,18)(21,24)(22,23)(25,34)(26,33)(27,36)(28,35)(29,32)(30,31) );
G=PermutationGroup([[(1,16,7),(2,13,8),(3,14,5),(4,15,6),(9,29,18),(10,30,19),(11,31,20),(12,32,17),(21,27,33),(22,28,34),(23,25,35),(24,26,36)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,35,11),(6,36,12),(7,33,9),(8,34,10),(13,28,19),(14,25,20),(15,26,17),(16,27,18)], [(1,33,27),(2,34,28),(3,35,25),(4,36,26),(5,14,31),(6,15,32),(7,16,29),(8,13,30),(9,18,21),(10,19,22),(11,20,23),(12,17,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36)], [(1,4),(2,3),(5,13),(6,16),(7,15),(8,14),(9,17),(10,20),(11,19),(12,18),(21,24),(22,23),(25,34),(26,33),(27,36),(28,35),(29,32),(30,31)]])
He3⋊5D4 is a maximal subgroup of
He3⋊3D8 He3⋊4SD16 He3⋊7SD16 He3⋊5D8 He3⋊7D8 He3⋊11SD16 C12.S32 C3⋊S3⋊D12 C62.47D6 D4×He3⋊C2 He3⋊5D4⋊C2
He3⋊5D4 is a maximal quotient of
He3⋊7SD16 He3⋊5D8 He3⋊5Q16 C62.30D6 C62.31D6
31 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 12A | 12B | 12C | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 18 | 18 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 2 | 2 | 6 | ··· | 6 |
31 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 6 |
| type | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | S3 | D4 | D6 | D12 | He3⋊C2 | C2×He3⋊C2 | He3⋊5D4 |
| kernel | He3⋊5D4 | C4×He3 | C2×He3⋊C2 | C3×C12 | He3 | C3×C6 | C32 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 1 | 4 | 8 | 4 | 4 | 2 |
Matrix representation of He3⋊5D4 ►in GL5(𝔽13)
| 0 | 12 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 12 | 8 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 0 | 12 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 11 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 4 | 3 | 3 |
| 3 | 7 | 0 | 0 | 0 |
| 6 | 10 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 6 | 10 | 0 | 0 | 0 |
| 3 | 7 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 8 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(13))| [0,1,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,8,1],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[0,1,0,0,0,12,12,0,0,0,0,0,10,9,4,0,0,10,0,3,0,0,11,0,3],[3,6,0,0,0,7,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[6,3,0,0,0,10,7,0,0,0,0,0,1,12,0,0,0,0,12,0,0,0,0,8,1] >;
He3⋊5D4 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_5D_4 % in TeX
G:=Group("He3:5D4"); // GroupNames label
G:=SmallGroup(216,68);
// by ID
G=gap.SmallGroup(216,68);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,387,1444,382]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^2=1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations