direct product, non-abelian, soluble, monomial
Aliases: C9×S4, A4⋊C18, (C3×S4).C3, C22⋊(S3×C9), (C9×A4)⋊1C2, (C2×C18)⋊1S3, C3.4(C3×S4), (C3×A4).2C6, (C2×C6).1(C3×S3), SmallGroup(216,89)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — C9×S4 |
Generators and relations for C9×S4
G = < a,b,c,d,e | a9=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >
Smallest permutation representation of C9×S4
►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 34)(11 35)(12 36)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 28)(7 29)(8 30)(9 31)(10 25)(11 26)(12 27)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(10 25 34)(11 26 35)(12 27 36)(13 19 28)(14 20 29)(15 21 30)(16 22 31)(17 23 32)(18 24 33)
(10 25)(11 26)(12 27)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
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)(10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33), (1,32)(2,33)(3,34)(4,35)(5,36)(6,28)(7,29)(8,30)(9,31)(10,25)(11,26)(12,27)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (10,25,34)(11,26,35)(12,27,36)(13,19,28)(14,20,29)(15,21,30)(16,22,31)(17,23,32)(18,24,33), (10,25)(11,26)(12,27)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)>;
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)(10,34)(11,35)(12,36)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33), (1,32)(2,33)(3,34)(4,35)(5,36)(6,28)(7,29)(8,30)(9,31)(10,25)(11,26)(12,27)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (10,25,34)(11,26,35)(12,27,36)(13,19,28)(14,20,29)(15,21,30)(16,22,31)(17,23,32)(18,24,33), (10,25)(11,26)(12,27)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24) );
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),(10,34),(11,35),(12,36),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,28),(7,29),(8,30),(9,31),(10,25),(11,26),(12,27),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(10,25,34),(11,26,35),(12,27,36),(13,19,28),(14,20,29),(15,21,30),(16,22,31),(17,23,32),(18,24,33)], [(10,25),(11,26),(12,27),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4 | 6A | 6B | 6C | 6D | 9A | ··· | 9F | 9G | ··· | 9L | 12A | 12B | 18A | ··· | 18F | 18G | ··· | 18L | 36A | ··· | 36F |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 3 | 6 | 1 | 1 | 8 | 8 | 8 | 6 | 3 | 3 | 6 | 6 | 1 | ··· | 1 | 8 | ··· | 8 | 6 | 6 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | S3 | C3×S3 | S3×C9 | S4 | C3×S4 | C9×S4 |
| kernel | C9×S4 | C9×A4 | C3×S4 | C3×A4 | S4 | A4 | C2×C18 | C2×C6 | C22 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 2 | 6 | 2 | 4 | 12 |
Matrix representation of C9×S4 ►in GL5(𝔽37)
| 34 | 0 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 | 0 |
| 0 | 0 | 26 | 0 | 0 |
| 0 | 0 | 0 | 26 | 0 |
| 0 | 0 | 0 | 0 | 26 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 1 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 1 | 36 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 36 |
| 0 | 0 | 1 | 0 | 36 |
| 0 | 0 | 0 | 0 | 36 |
| 36 | 36 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 36 | 36 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(37))| [34,0,0,0,0,0,34,0,0,0,0,0,26,0,0,0,0,0,26,0,0,0,0,0,26],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,36,36,36,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,36,36,36],[36,1,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[36,0,0,0,0,36,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;
C9×S4 in GAP, Magma, Sage, TeX
C_9\times S_4
% in TeX
G:=Group("C9xS4"); // GroupNames label
G:=SmallGroup(216,89);
// by ID
G=gap.SmallGroup(216,89);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-2,2,43,867,3244,202,1949,347]);
// Polycyclic
G:=Group<a,b,c,d,e|a^9=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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