metabelian, soluble, monomial, A-group
Aliases: C4.3F9, C3⋊S3⋊C16, C2.F9⋊3C2, C2.1(C2×F9), (C3×C12).2C8, C32⋊1(C2×C16), C32⋊2C8.1C4, C32⋊2C8.3C22, (C4×C3⋊S3).1C4, (C2×C3⋊S3).1C8, (C3×C6).1(C2×C8), C3⋊S3⋊3C8.8C2, C3⋊Dic3.1(C2×C4), SmallGroup(288,861)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C4.3F9 |
Generators and relations for C4.3F9
G = < a,b,c,d | a4=b3=c3=1, d8=a2, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of C4.3F9
►On 48 points
Generators in S48
(1 13 9 5)(2 14 10 6)(3 15 11 7)(4 16 12 8)(17 34 25 42)(18 35 26 43)(19 36 27 44)(20 37 28 45)(21 38 29 46)(22 39 30 47)(23 40 31 48)(24 41 32 33)
(2 21 42)(3 22 43)(4 44 23)(6 46 25)(7 47 26)(8 27 48)(10 29 34)(11 30 35)(12 36 31)(14 38 17)(15 39 18)(16 19 40)
(1 20 41)(3 22 43)(4 23 44)(5 45 24)(7 47 26)(8 48 27)(9 28 33)(11 30 35)(12 31 36)(13 37 32)(15 39 18)(16 40 19)
(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)
G:=sub<Sym(48)| (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,34,25,42)(18,35,26,43)(19,36,27,44)(20,37,28,45)(21,38,29,46)(22,39,30,47)(23,40,31,48)(24,41,32,33), (2,21,42)(3,22,43)(4,44,23)(6,46,25)(7,47,26)(8,27,48)(10,29,34)(11,30,35)(12,36,31)(14,38,17)(15,39,18)(16,19,40), (1,20,41)(3,22,43)(4,23,44)(5,45,24)(7,47,26)(8,48,27)(9,28,33)(11,30,35)(12,31,36)(13,37,32)(15,39,18)(16,40,19), (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)>;
G:=Group( (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,34,25,42)(18,35,26,43)(19,36,27,44)(20,37,28,45)(21,38,29,46)(22,39,30,47)(23,40,31,48)(24,41,32,33), (2,21,42)(3,22,43)(4,44,23)(6,46,25)(7,47,26)(8,27,48)(10,29,34)(11,30,35)(12,36,31)(14,38,17)(15,39,18)(16,19,40), (1,20,41)(3,22,43)(4,23,44)(5,45,24)(7,47,26)(8,48,27)(9,28,33)(11,30,35)(12,31,36)(13,37,32)(15,39,18)(16,40,19), (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) );
G=PermutationGroup([(1,13,9,5),(2,14,10,6),(3,15,11,7),(4,16,12,8),(17,34,25,42),(18,35,26,43),(19,36,27,44),(20,37,28,45),(21,38,29,46),(22,39,30,47),(23,40,31,48),(24,41,32,33)], [(2,21,42),(3,22,43),(4,44,23),(6,46,25),(7,47,26),(8,27,48),(10,29,34),(11,30,35),(12,36,31),(14,38,17),(15,39,18),(16,19,40)], [(1,20,41),(3,22,43),(4,23,44),(5,45,24),(7,47,26),(8,48,27),(9,28,33),(11,30,35),(12,31,36),(13,37,32),(15,39,18),(16,40,19)], [(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)])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 8A | ··· | 8H | 12A | 12B | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 8 | ··· | 8 | 12 | 12 | 16 | ··· | 16 |
| size | 1 | 1 | 9 | 9 | 8 | 1 | 1 | 9 | 9 | 8 | 9 | ··· | 9 | 8 | 8 | 9 | ··· | 9 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 8 | 8 |
| type | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | F9 | C2×F9 | C4.3F9 |
| kernel | C4.3F9 | C2.F9 | C3⋊S3⋊3C8 | C32⋊2C8 | C4×C3⋊S3 | C3×C12 | C2×C3⋊S3 | C3⋊S3 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 1 | 1 | 2 |
Matrix representation of C4.3F9 ►in GL9(𝔽97)
| 75 | 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 | 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 |
| 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 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 96 | 96 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 96 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 | 0 | 0 | 0 | 96 | 96 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 96 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 96 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 96 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 96 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 96 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 96 | 96 | 0 | 0 |
| 0 | 0 | 96 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 96 | 0 | 0 | 0 | 0 | 0 | 1 |
| 70 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 96 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 95 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 96 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 96 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 96 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 96 | 0 | 0 | 0 |
G:=sub<GL(9,GF(97))| [75,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0,0,0,0,96,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,96,1,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,96],[1,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,96,96,96,0,96,0,96,96,0,0,0,0,96,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[70,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,96,95,96,96,96,96,96,96,0,1,96,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
C4.3F9 in GAP, Magma, Sage, TeX
C_4._3F_9
% in TeX
G:=Group("C4.3F9"); // GroupNames label
G:=SmallGroup(288,861);
// by ID
G=gap.SmallGroup(288,861);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,120,58,80,4037,2371,362,10982,3156,1203]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^3=c^3=1,d^8=a^2,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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