metabelian, soluble, monomial, A-group
Aliases: C9.A4, C22⋊C27, (C2×C6).C9, (C2×C18).C3, C3.(C3.A4), SmallGroup(108,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C9.A4 |
Generators and relations for C9.A4
G = < a,b,c,d | a9=b2=c2=1, d3=a, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of C9.A4
►On 54 points
Generators in S54
(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)(28 31 34 37 40 43 46 49 52)(29 32 35 38 41 44 47 50 53)(30 33 36 39 42 45 48 51 54)
(2 45)(3 46)(5 48)(6 49)(8 51)(9 52)(11 54)(12 28)(14 30)(15 31)(17 33)(18 34)(20 36)(21 37)(23 39)(24 40)(26 42)(27 43)
(1 44)(3 46)(4 47)(6 49)(7 50)(9 52)(10 53)(12 28)(13 29)(15 31)(16 32)(18 34)(19 35)(21 37)(22 38)(24 40)(25 41)(27 43)
(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 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)
G:=sub<Sym(54)| (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)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54), (2,45)(3,46)(5,48)(6,49)(8,51)(9,52)(11,54)(12,28)(14,30)(15,31)(17,33)(18,34)(20,36)(21,37)(23,39)(24,40)(26,42)(27,43), (1,44)(3,46)(4,47)(6,49)(7,50)(9,52)(10,53)(12,28)(13,29)(15,31)(16,32)(18,34)(19,35)(21,37)(22,38)(24,40)(25,41)(27,43), (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)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)>;
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)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54), (2,45)(3,46)(5,48)(6,49)(8,51)(9,52)(11,54)(12,28)(14,30)(15,31)(17,33)(18,34)(20,36)(21,37)(23,39)(24,40)(26,42)(27,43), (1,44)(3,46)(4,47)(6,49)(7,50)(9,52)(10,53)(12,28)(13,29)(15,31)(16,32)(18,34)(19,35)(21,37)(22,38)(24,40)(25,41)(27,43), (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)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54) );
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),(28,31,34,37,40,43,46,49,52),(29,32,35,38,41,44,47,50,53),(30,33,36,39,42,45,48,51,54)], [(2,45),(3,46),(5,48),(6,49),(8,51),(9,52),(11,54),(12,28),(14,30),(15,31),(17,33),(18,34),(20,36),(21,37),(23,39),(24,40),(26,42),(27,43)], [(1,44),(3,46),(4,47),(6,49),(7,50),(9,52),(10,53),(12,28),(13,29),(15,31),(16,32),(18,34),(19,35),(21,37),(22,38),(24,40),(25,41),(27,43)], [(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),(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)]])
C9.A4 is a maximal subgroup of
C9.S4 A4×C27 C27⋊A4 C62.C9 C42⋊C27 C24⋊C27
C9.A4 is a maximal quotient of Q8⋊C27 C27.A4 C42⋊C27 C24⋊C27
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 9A | ··· | 9F | 18A | ··· | 18F | 27A | ··· | 27R |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | + | |||||
| image | C1 | C3 | C9 | C27 | A4 | C3.A4 | C9.A4 |
| kernel | C9.A4 | C2×C18 | C2×C6 | C22 | C9 | C3 | C1 |
| # reps | 1 | 2 | 6 | 18 | 1 | 2 | 6 |
Matrix representation of C9.A4 ►in GL3(𝔽109) generated by
| 27 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 0 | 27 |
| 1 | 0 | 0 |
| 22 | 108 | 0 |
| 27 | 0 | 108 |
| 108 | 0 | 0 |
| 0 | 108 | 0 |
| 82 | 0 | 1 |
| 22 | 107 | 0 |
| 65 | 87 | 1 |
| 11 | 82 | 0 |
G:=sub<GL(3,GF(109))| [27,0,0,0,27,0,0,0,27],[1,22,27,0,108,0,0,0,108],[108,0,82,0,108,0,0,0,1],[22,65,11,107,87,82,0,1,0] >;
C9.A4 in GAP, Magma, Sage, TeX
C_9.A_4
% in TeX
G:=Group("C9.A4"); // GroupNames label
G:=SmallGroup(108,3);
// by ID
G=gap.SmallGroup(108,3);
# by ID
G:=PCGroup([5,-3,-3,-3,-2,2,15,36,1083,2029]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^2=c^2=1,d^3=a,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
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