p-group, metabelian, nilpotent (class 3), monomial
Aliases: C9.4He3, C33.2C9, C9.43- 1+2, C27⋊C3⋊1C3, (C3×C9).3C9, (C32×C9).9C3, C3.6(C32⋊C9), C32.10(C3×C9), (C3×C9).24C32, SmallGroup(243,16)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C9.4He3
G = < a,b,c,d | a9=b3=c3=1, d3=a, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a3bc-1, dcd-1=a6c >
Permutation representations of C9.4He3
►On 27 points - transitive group 27T92
Generators in S27
(1 4 7 10 13 16 19 22 25)(2 5 8 11 14 17 20 23 26)(3 6 9 12 15 18 21 24 27)
(1 10 19)(2 20 11)(3 12 21)(4 13 22)(5 23 14)(6 15 24)(7 16 25)(8 26 17)(9 18 27)
(2 20 11)(3 12 21)(5 23 14)(6 15 24)(8 26 17)(9 18 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)
G:=sub<Sym(27)| (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27), (1,10,19)(2,20,11)(3,12,21)(4,13,22)(5,23,14)(6,15,24)(7,16,25)(8,26,17)(9,18,27), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27), (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)>;
G:=Group( (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27), (1,10,19)(2,20,11)(3,12,21)(4,13,22)(5,23,14)(6,15,24)(7,16,25)(8,26,17)(9,18,27), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27), (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) );
G=PermutationGroup([[(1,4,7,10,13,16,19,22,25),(2,5,8,11,14,17,20,23,26),(3,6,9,12,15,18,21,24,27)], [(1,10,19),(2,20,11),(3,12,21),(4,13,22),(5,23,14),(6,15,24),(7,16,25),(8,26,17),(9,18,27)], [(2,20,11),(3,12,21),(5,23,14),(6,15,24),(8,26,17),(9,18,27)], [(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)]])
G:=TransitiveGroup(27,92);
C9.4He3 is a maximal subgroup of
C33.D9
51 conjugacy classes
| class | 1 | 3A | 3B | 3C | ··· | 3J | 9A | ··· | 9F | 9G | ··· | 9V | 27A | ··· | 27R |
| order | 1 | 3 | 3 | 3 | ··· | 3 | 9 | ··· | 9 | 9 | ··· | 9 | 27 | ··· | 27 |
| size | 1 | 1 | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | |||||||
| image | C1 | C3 | C3 | C9 | C9 | He3 | 3- 1+2 | C9.4He3 |
| kernel | C9.4He3 | C27⋊C3 | C32×C9 | C3×C9 | C33 | C9 | C9 | C1 |
| # reps | 1 | 6 | 2 | 12 | 6 | 2 | 4 | 18 |
Matrix representation of C9.4He3 ►in GL3(𝔽109) generated by
| 66 | 0 | 0 |
| 0 | 66 | 0 |
| 0 | 0 | 66 |
| 45 | 0 | 0 |
| 0 | 45 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 63 | 0 |
| 0 | 0 | 45 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 66 | 0 | 0 |
G:=sub<GL(3,GF(109))| [66,0,0,0,66,0,0,0,66],[45,0,0,0,45,0,0,0,1],[1,0,0,0,63,0,0,0,45],[0,0,66,1,0,0,0,1,0] >;
C9.4He3 in GAP, Magma, Sage, TeX
C_9._4{\rm He}_3 % in TeX
G:=Group("C9.4He3"); // GroupNames label
G:=SmallGroup(243,16);
// by ID
G=gap.SmallGroup(243,16);
# by ID
G:=PCGroup([5,-3,3,-3,3,-3,135,121,1352,78]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^3=c^3=1,d^3=a,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b*c^-1,d*c*d^-1=a^6*c>;
// generators/relations
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