non-abelian, supersoluble, monomial
Aliases: He3⋊7D4, C62⋊3S3, (C3×C6).20D6, He3⋊3C4⋊3C2, C32⋊5(C3⋊D4), (C22×He3)⋊3C2, C3.2(C32⋊7D4), C22⋊2(He3⋊C2), (C2×He3).15C22, C6.29(C2×C3⋊S3), (C2×C6).4(C3⋊S3), (C2×He3⋊C2)⋊3C2, C2.5(C2×He3⋊C2), SmallGroup(216,72)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊7D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, cac-1=ab-1, dad-1=eae=a-1, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 310 in 88 conjugacy classes, 24 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, D4, C32, Dic3, C12, D6, C2×C6, C2×C6, C3×S3, C3×C6, C3×C6, C3⋊D4, C3×D4, He3, C3×Dic3, S3×C6, C62, He3⋊C2, C2×He3, C2×He3, C3×C3⋊D4, He3⋊3C4, C2×He3⋊C2, C22×He3, He3⋊7D4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊S3, C3⋊D4, C2×C3⋊S3, He3⋊C2, C32⋊7D4, C2×He3⋊C2, He3⋊7D4
►On 36 points
(1 35 22)(2 23 36)(3 33 24)(4 21 34)(5 11 16)(6 13 12)(7 9 14)(8 15 10)(17 26 31)(18 32 27)(19 28 29)(20 30 25)
(1 30 5)(2 31 6)(3 32 7)(4 29 8)(9 33 27)(10 34 28)(11 35 25)(12 36 26)(13 23 17)(14 24 18)(15 21 19)(16 22 20)
(9 33 27)(10 28 34)(11 35 25)(12 26 36)(13 23 17)(14 18 24)(15 21 19)(16 20 22)
(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)
(2 4)(6 8)(9 14)(10 13)(11 16)(12 15)(17 28)(18 27)(19 26)(20 25)(21 36)(22 35)(23 34)(24 33)(29 31)
G:=sub<Sym(36)| (1,35,22)(2,23,36)(3,33,24)(4,21,34)(5,11,16)(6,13,12)(7,9,14)(8,15,10)(17,26,31)(18,32,27)(19,28,29)(20,30,25), (1,30,5)(2,31,6)(3,32,7)(4,29,8)(9,33,27)(10,34,28)(11,35,25)(12,36,26)(13,23,17)(14,24,18)(15,21,19)(16,22,20), (9,33,27)(10,28,34)(11,35,25)(12,26,36)(13,23,17)(14,18,24)(15,21,19)(16,20,22), (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), (2,4)(6,8)(9,14)(10,13)(11,16)(12,15)(17,28)(18,27)(19,26)(20,25)(21,36)(22,35)(23,34)(24,33)(29,31)>;
G:=Group( (1,35,22)(2,23,36)(3,33,24)(4,21,34)(5,11,16)(6,13,12)(7,9,14)(8,15,10)(17,26,31)(18,32,27)(19,28,29)(20,30,25), (1,30,5)(2,31,6)(3,32,7)(4,29,8)(9,33,27)(10,34,28)(11,35,25)(12,36,26)(13,23,17)(14,24,18)(15,21,19)(16,22,20), (9,33,27)(10,28,34)(11,35,25)(12,26,36)(13,23,17)(14,18,24)(15,21,19)(16,20,22), (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), (2,4)(6,8)(9,14)(10,13)(11,16)(12,15)(17,28)(18,27)(19,26)(20,25)(21,36)(22,35)(23,34)(24,33)(29,31) );
G=PermutationGroup([[(1,35,22),(2,23,36),(3,33,24),(4,21,34),(5,11,16),(6,13,12),(7,9,14),(8,15,10),(17,26,31),(18,32,27),(19,28,29),(20,30,25)], [(1,30,5),(2,31,6),(3,32,7),(4,29,8),(9,33,27),(10,34,28),(11,35,25),(12,36,26),(13,23,17),(14,24,18),(15,21,19),(16,22,20)], [(9,33,27),(10,28,34),(11,35,25),(12,26,36),(13,23,17),(14,18,24),(15,21,19),(16,20,22)], [(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)], [(2,4),(6,8),(9,14),(10,13),(11,16),(12,15),(17,28),(18,27),(19,26),(20,25),(21,36),(22,35),(23,34),(24,33),(29,31)]])
He3⋊7D4 is a maximal subgroup of
C62.8D6 C62⋊D6 C62.47D6 D4×He3⋊C2 C62.16D6
He3⋊7D4 is a maximal quotient of C62.29D6 C62.31D6 He3⋊7D8 He3⋊9SD16 He3⋊11SD16 He3⋊7Q16 C62⋊4Dic3
31 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4 | 6A | 6B | 6C | 6D | 6E | ··· | 6P | 6Q | 6R | 12A | 12B |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 12 | 12 |
| size | 1 | 1 | 2 | 18 | 1 | 1 | 6 | 6 | 6 | 6 | 18 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 18 | 18 | 18 | 18 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 6 |
| type | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | C3⋊D4 | He3⋊C2 | C2×He3⋊C2 | He3⋊7D4 |
| kernel | He3⋊7D4 | He3⋊3C4 | C2×He3⋊C2 | C22×He3 | C62 | He3 | C3×C6 | C32 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 8 | 4 | 4 | 2 |
Matrix representation of He3⋊7D4 ►in GL5(𝔽13)
| 0 | 1 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 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 |
| 12 | 12 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 11 | 2 | 0 | 0 | 0 |
| 4 | 2 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(13))| [0,12,0,0,0,1,12,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[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],[12,1,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,9],[11,4,0,0,0,2,2,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,1,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
He3⋊7D4 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_7D_4 % in TeX
G:=Group("He3:7D4"); // GroupNames label
G:=SmallGroup(216,72);
// by ID
G=gap.SmallGroup(216,72);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,387,1444,382]);
// 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,d*a*d^-1=e*a*e=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations