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 40 25 48)(18 41 26 33)(19 42 27 34)(20 43 28 35)(21 44 29 36)(22 45 30 37)(23 46 31 38)(24 47 32 39)
(2 27 38)(3 28 39)(4 40 29)(6 42 31)(7 43 32)(8 17 44)(10 19 46)(11 20 47)(12 48 21)(14 34 23)(15 35 24)(16 25 36)
(1 26 37)(3 28 39)(4 29 40)(5 41 30)(7 43 32)(8 44 17)(9 18 45)(11 20 47)(12 21 48)(13 33 22)(15 35 24)(16 36 25)
(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)(17,40,25,48)(18,41,26,33)(19,42,27,34)(20,43,28,35)(21,44,29,36)(22,45,30,37)(23,46,31,38)(24,47,32,39), (2,27,38)(3,28,39)(4,40,29)(6,42,31)(7,43,32)(8,17,44)(10,19,46)(11,20,47)(12,48,21)(14,34,23)(15,35,24)(16,25,36), (1,26,37)(3,28,39)(4,29,40)(5,41,30)(7,43,32)(8,44,17)(9,18,45)(11,20,47)(12,21,48)(13,33,22)(15,35,24)(16,36,25), (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)(17,40,25,48)(18,41,26,33)(19,42,27,34)(20,43,28,35)(21,44,29,36)(22,45,30,37)(23,46,31,38)(24,47,32,39), (2,27,38)(3,28,39)(4,40,29)(6,42,31)(7,43,32)(8,17,44)(10,19,46)(11,20,47)(12,48,21)(14,34,23)(15,35,24)(16,25,36), (1,26,37)(3,28,39)(4,29,40)(5,41,30)(7,43,32)(8,44,17)(9,18,45)(11,20,47)(12,21,48)(13,33,22)(15,35,24)(16,36,25), (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),(17,40,25,48),(18,41,26,33),(19,42,27,34),(20,43,28,35),(21,44,29,36),(22,45,30,37),(23,46,31,38),(24,47,32,39)], [(2,27,38),(3,28,39),(4,40,29),(6,42,31),(7,43,32),(8,17,44),(10,19,46),(11,20,47),(12,48,21),(14,34,23),(15,35,24),(16,25,36)], [(1,26,37),(3,28,39),(4,29,40),(5,41,30),(7,43,32),(8,44,17),(9,18,45),(11,20,47),(12,21,48),(13,33,22),(15,35,24),(16,36,25)], [(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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