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 [×2], C3, C3 [×4], C4, C22 [×2], S3 [×8], C6, C6 [×6], D4, C32 [×4], C12, C12 [×4], D6 [×8], C2×C6 [×2], C3×S3 [×8], C3×C6 [×4], D12 [×4], C3×D4, He3, C3×C12 [×4], S3×C6 [×8], He3⋊C2 [×2], C2×He3, C3×D12 [×4], C4×He3, C2×He3⋊C2 [×2], He3⋊5D4
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], C3⋊S3, D12 [×4], C2×C3⋊S3, He3⋊C2, C12⋊S3, C2×He3⋊C2, He3⋊5D4
►On 36 points
(1 16 24)(2 13 21)(3 14 22)(4 15 23)(5 25 35)(6 26 36)(7 27 33)(8 28 34)(9 18 29)(10 19 30)(11 20 31)(12 17 32)
(1 7 29)(2 8 30)(3 5 31)(4 6 32)(9 16 27)(10 13 28)(11 14 25)(12 15 26)(17 23 36)(18 24 33)(19 21 34)(20 22 35)
(1 33 27)(2 34 28)(3 35 25)(4 36 26)(5 20 11)(6 17 12)(7 18 9)(8 19 10)(13 30 21)(14 31 22)(15 32 23)(16 29 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 8)(6 7)(9 17)(10 20)(11 19)(12 18)(13 22)(14 21)(15 24)(16 23)(25 34)(26 33)(27 36)(28 35)(29 32)(30 31)
G:=sub<Sym(36)| (1,16,24)(2,13,21)(3,14,22)(4,15,23)(5,25,35)(6,26,36)(7,27,33)(8,28,34)(9,18,29)(10,19,30)(11,20,31)(12,17,32), (1,7,29)(2,8,30)(3,5,31)(4,6,32)(9,16,27)(10,13,28)(11,14,25)(12,15,26)(17,23,36)(18,24,33)(19,21,34)(20,22,35), (1,33,27)(2,34,28)(3,35,25)(4,36,26)(5,20,11)(6,17,12)(7,18,9)(8,19,10)(13,30,21)(14,31,22)(15,32,23)(16,29,24), (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)(5,8)(6,7)(9,17)(10,20)(11,19)(12,18)(13,22)(14,21)(15,24)(16,23)(25,34)(26,33)(27,36)(28,35)(29,32)(30,31)>;
G:=Group( (1,16,24)(2,13,21)(3,14,22)(4,15,23)(5,25,35)(6,26,36)(7,27,33)(8,28,34)(9,18,29)(10,19,30)(11,20,31)(12,17,32), (1,7,29)(2,8,30)(3,5,31)(4,6,32)(9,16,27)(10,13,28)(11,14,25)(12,15,26)(17,23,36)(18,24,33)(19,21,34)(20,22,35), (1,33,27)(2,34,28)(3,35,25)(4,36,26)(5,20,11)(6,17,12)(7,18,9)(8,19,10)(13,30,21)(14,31,22)(15,32,23)(16,29,24), (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)(5,8)(6,7)(9,17)(10,20)(11,19)(12,18)(13,22)(14,21)(15,24)(16,23)(25,34)(26,33)(27,36)(28,35)(29,32)(30,31) );
G=PermutationGroup([(1,16,24),(2,13,21),(3,14,22),(4,15,23),(5,25,35),(6,26,36),(7,27,33),(8,28,34),(9,18,29),(10,19,30),(11,20,31),(12,17,32)], [(1,7,29),(2,8,30),(3,5,31),(4,6,32),(9,16,27),(10,13,28),(11,14,25),(12,15,26),(17,23,36),(18,24,33),(19,21,34),(20,22,35)], [(1,33,27),(2,34,28),(3,35,25),(4,36,26),(5,20,11),(6,17,12),(7,18,9),(8,19,10),(13,30,21),(14,31,22),(15,32,23),(16,29,24)], [(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),(5,8),(6,7),(9,17),(10,20),(11,19),(12,18),(13,22),(14,21),(15,24),(16,23),(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