direct product, metabelian, soluble, monomial, A-group
Aliases: C4×F9, C32⋊(C4×C8), (C3×C12)⋊1C8, C32⋊C4⋊1C8, C2.2(C2×F9), C3⋊Dic3⋊1C8, (C2×F9).4C2, C3⋊S3.1C42, (C4×C3⋊S3).3C4, C3⋊S3.1(C2×C8), (C3×C6).3(C2×C8), C32⋊C4.4(C2×C4), (C2×C32⋊C4).4C4, (C4×C32⋊C4).6C2, (C2×C32⋊C4).9C22, (C2×C3⋊S3).1(C2×C4), SmallGroup(288,863)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C4×F9 |
Generators and relations for C4×F9
G = < a,b,c,d | a4=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of C4×F9
►On 36 points
Generators in S36
(1 3 2 4)(5 32 19 21)(6 33 20 22)(7 34 13 23)(8 35 14 24)(9 36 15 25)(10 29 16 26)(11 30 17 27)(12 31 18 28)
(1 6 10)(2 20 16)(3 33 29)(4 22 26)(5 12 7)(8 9 11)(13 19 18)(14 15 17)(21 28 23)(24 25 27)(30 35 36)(31 34 32)
(1 7 11)(2 13 17)(3 34 30)(4 23 27)(5 8 6)(9 10 12)(14 20 19)(15 16 18)(21 24 22)(25 26 28)(29 31 36)(32 35 33)
(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)
G:=sub<Sym(36)| (1,3,2,4)(5,32,19,21)(6,33,20,22)(7,34,13,23)(8,35,14,24)(9,36,15,25)(10,29,16,26)(11,30,17,27)(12,31,18,28), (1,6,10)(2,20,16)(3,33,29)(4,22,26)(5,12,7)(8,9,11)(13,19,18)(14,15,17)(21,28,23)(24,25,27)(30,35,36)(31,34,32), (1,7,11)(2,13,17)(3,34,30)(4,23,27)(5,8,6)(9,10,12)(14,20,19)(15,16,18)(21,24,22)(25,26,28)(29,31,36)(32,35,33), (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)>;
G:=Group( (1,3,2,4)(5,32,19,21)(6,33,20,22)(7,34,13,23)(8,35,14,24)(9,36,15,25)(10,29,16,26)(11,30,17,27)(12,31,18,28), (1,6,10)(2,20,16)(3,33,29)(4,22,26)(5,12,7)(8,9,11)(13,19,18)(14,15,17)(21,28,23)(24,25,27)(30,35,36)(31,34,32), (1,7,11)(2,13,17)(3,34,30)(4,23,27)(5,8,6)(9,10,12)(14,20,19)(15,16,18)(21,24,22)(25,26,28)(29,31,36)(32,35,33), (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) );
G=PermutationGroup([[(1,3,2,4),(5,32,19,21),(6,33,20,22),(7,34,13,23),(8,35,14,24),(9,36,15,25),(10,29,16,26),(11,30,17,27),(12,31,18,28)], [(1,6,10),(2,20,16),(3,33,29),(4,22,26),(5,12,7),(8,9,11),(13,19,18),(14,15,17),(21,28,23),(24,25,27),(30,35,36),(31,34,32)], [(1,7,11),(2,13,17),(3,34,30),(4,23,27),(5,8,6),(9,10,12),(14,20,19),(15,16,18),(21,24,22),(25,26,28),(29,31,36),(32,35,33)], [(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)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | ··· | 4L | 6 | 8A | ··· | 8P | 12A | 12B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 6 | 8 | ··· | 8 | 12 | 12 |
| size | 1 | 1 | 9 | 9 | 8 | 1 | 1 | 9 | ··· | 9 | 8 | 9 | ··· | 9 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 8 | 8 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | F9 | C2×F9 | C4×F9 |
| kernel | C4×F9 | C4×C32⋊C4 | C2×F9 | C4×C3⋊S3 | F9 | C2×C32⋊C4 | C3⋊Dic3 | C3×C12 | C32⋊C4 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 8 | 2 | 4 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of C4×F9 ►in GL9(𝔽73)
| 27 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 1 | 0 |
| 51 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(9,GF(73))| [27,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,72,72,72,72,72,72,72,72,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,72,72,72,72,72,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[51,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0] >;
C4×F9 in GAP, Magma, Sage, TeX
C_4\times F_9
% in TeX
G:=Group("C4xF9"); // GroupNames label
G:=SmallGroup(288,863);
// by ID
G=gap.SmallGroup(288,863);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,64,100,4037,2371,201,10982,3156,622]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^3=c^3=d^8=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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