direct product, metabelian, soluble, monomial, A-group
Aliases: C9×A4, (C2×C18)⋊1C3, C9○(C3.A4), C3.A4⋊3C3, C3.1(C3×A4), C22⋊1(C3×C9), (C3×A4).2C3, (C2×C6).1C32, SmallGroup(108,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C9×A4 |
Generators and relations for C9×A4
G = < a,b,c,d | a9=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of C9×A4
►On 36 points
Generators in S36
(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 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 28)(18 29)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 10)(9 11)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)
(1 7 4)(2 8 5)(3 9 6)(10 36 24)(11 28 25)(12 29 26)(13 30 27)(14 31 19)(15 32 20)(16 33 21)(17 34 22)(18 35 23)
G:=sub<Sym(36)| (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), (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,10)(9,11)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (1,7,4)(2,8,5)(3,9,6)(10,36,24)(11,28,25)(12,29,26)(13,30,27)(14,31,19)(15,32,20)(16,33,21)(17,34,22)(18,35,23)>;
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)(28,29,30,31,32,33,34,35,36), (1,23)(2,24)(3,25)(4,26)(5,27)(6,19)(7,20)(8,21)(9,22)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,10)(9,11)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (1,7,4)(2,8,5)(3,9,6)(10,36,24)(11,28,25)(12,29,26)(13,30,27)(14,31,19)(15,32,20)(16,33,21)(17,34,22)(18,35,23) );
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),(28,29,30,31,32,33,34,35,36)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,19),(7,20),(8,21),(9,22),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,28),(18,29)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,10),(9,11),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36)], [(1,7,4),(2,8,5),(3,9,6),(10,36,24),(11,28,25),(12,29,26),(13,30,27),(14,31,19),(15,32,20),(16,33,21),(17,34,22),(18,35,23)]])
C9×A4 is a maximal subgroup of
C9⋊S4 C27⋊A4 C62.25C32 C62.9C32 C3.A42
C9×A4 is a maximal quotient of C27⋊A4 C62.11C32 C62.16C32 C3.A42
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3H | 6A | 6B | 9A | ··· | 9F | 9G | ··· | 9R | 18A | ··· | 18F |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 3 | 1 | 1 | 4 | ··· | 4 | 3 | 3 | 1 | ··· | 1 | 4 | ··· | 4 | 3 | ··· | 3 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | + | ||||||
| image | C1 | C3 | C3 | C3 | C9 | A4 | C3×A4 | C9×A4 |
| kernel | C9×A4 | C3.A4 | C2×C18 | C3×A4 | A4 | C9 | C3 | C1 |
| # reps | 1 | 4 | 2 | 2 | 18 | 1 | 2 | 6 |
Matrix representation of C9×A4 ►in GL3(𝔽19) generated by
| 9 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 0 | 18 | 1 |
| 0 | 18 | 0 |
| 1 | 18 | 0 |
| 18 | 0 | 0 |
| 18 | 0 | 1 |
| 18 | 1 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 11 | 0 | 0 |
G:=sub<GL(3,GF(19))| [9,0,0,0,9,0,0,0,9],[0,0,1,18,18,18,1,0,0],[18,18,18,0,0,1,0,1,0],[0,0,11,11,0,0,0,11,0] >;
C9×A4 in GAP, Magma, Sage, TeX
C_9\times A_4
% in TeX
G:=Group("C9xA4"); // GroupNames label
G:=SmallGroup(108,18);
// by ID
G=gap.SmallGroup(108,18);
# by ID
G:=PCGroup([5,-3,-3,-3,-2,2,36,1083,2029]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^2=c^2=d^3=1,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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