p-group, metabelian, nilpotent (class 3), monomial
Aliases: C9.1He3, He3.2C32, C32.5C33, C33.35C32, 3- 1+2.2C32, C3≀C3⋊3C3, C9○He3⋊1C3, (C32×C9)⋊8C3, He3.C3⋊3C3, C3.11(C3×He3), He3⋊C3⋊5C3, C3.He3⋊6C3, (C3×C9).29C32, SmallGroup(243,55)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C9.He3
G = < a,b,c,d | a9=b3=c3=d3=1, ab=ba, ac=ca, dad-1=a4, bc=cb, dbd-1=a3bc-1, dcd-1=a6c >
Subgroups: 153 in 67 conjugacy classes, 33 normal (11 characteristic)
C1, C3, C3, C9, C9, C9, C32, C32, C3×C9, C3×C9, He3, 3- 1+2, 3- 1+2, C33, C3≀C3, He3.C3, He3⋊C3, C3.He3, C32×C9, C9○He3, C9.He3
Quotients: C1, C3, C32, He3, C33, C3×He3, C9.He3
►On 27 points - transitive group 27T93
(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)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)
(1 20 14)(2 27 18)(3 25 13)(4 23 17)(5 21 12)(6 19 16)(7 26 11)(8 24 15)(9 22 10)
G:=sub<Sym(27)| (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), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24), (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24), (1,20,14)(2,27,18)(3,25,13)(4,23,17)(5,21,12)(6,19,16)(7,26,11)(8,24,15)(9,22,10)>;
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), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24), (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24), (1,20,14)(2,27,18)(3,25,13)(4,23,17)(5,21,12)(6,19,16)(7,26,11)(8,24,15)(9,22,10) );
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)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24)], [(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24)], [(1,20,14),(2,27,18),(3,25,13),(4,23,17),(5,21,12),(6,19,16),(7,26,11),(8,24,15),(9,22,10)]])
G:=TransitiveGroup(27,93);
C9.He3 is a maximal subgroup of
C3≀S3⋊3C3 C3≀C3.S3 C3≀C3⋊S3
51 conjugacy classes
| class | 1 | 3A | 3B | 3C | ··· | 3J | 3K | ··· | 3P | 9A | ··· | 9F | 9G | ··· | 9V | 9W | ··· | 9AH |
| order | 1 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 9 | ··· | 9 | 9 | ··· | 9 | 9 | ··· | 9 |
| size | 1 | 1 | 1 | 3 | ··· | 3 | 9 | ··· | 9 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | ||||||||
| image | C1 | C3 | C3 | C3 | C3 | C3 | C3 | He3 | C9.He3 |
| kernel | C9.He3 | C3≀C3 | He3.C3 | He3⋊C3 | C3.He3 | C32×C9 | C9○He3 | C9 | C1 |
| # reps | 1 | 6 | 6 | 2 | 4 | 2 | 6 | 6 | 18 |
Matrix representation of C9.He3 ►in GL3(𝔽19) generated by
| 9 | 0 | 0 |
| 2 | 4 | 0 |
| 15 | 0 | 6 |
| 11 | 0 | 0 |
| 0 | 11 | 0 |
| 12 | 0 | 1 |
| 1 | 0 | 0 |
| 9 | 7 | 0 |
| 7 | 0 | 11 |
| 9 | 6 | 0 |
| 10 | 10 | 1 |
| 4 | 11 | 0 |
G:=sub<GL(3,GF(19))| [9,2,15,0,4,0,0,0,6],[11,0,12,0,11,0,0,0,1],[1,9,7,0,7,0,0,0,11],[9,10,4,6,10,11,0,1,0] >;
C9.He3 in GAP, Magma, Sage, TeX
C_9.{\rm He}_3 % in TeX
G:=Group("C9.He3"); // GroupNames label
G:=SmallGroup(243,55);
// by ID
G=gap.SmallGroup(243,55);
# by ID
G:=PCGroup([5,-3,3,3,-3,-3,301,147,2163]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^4,b*c=c*b,d*b*d^-1=a^3*b*c^-1,d*c*d^-1=a^6*c>;
// generators/relations