metabelian, soluble, monomial, A-group
Aliases: C5⋊F9, C32⋊(C5⋊C8), C3⋊S3.F5, (C3×C15)⋊3C8, C32⋊Dic5.1C2, (C5×C3⋊S3).3C4, SmallGroup(360,125)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — C5⋊F9 |
Generators and relations for C5⋊F9
G = < a,b,c,d | a5=b3=c3=d8=1, ab=ba, ac=ca, dad-1=a3, dbd-1=bc=cb, dcd-1=b >
Character table of C5⋊F9
| class | 1 | 2 | 3 | 4A | 4B | 5 | 8A | 8B | 8C | 8D | 10 | 15A | 15B | 15C | 15D | |
| size | 1 | 9 | 8 | 45 | 45 | 4 | 45 | 45 | 45 | 45 | 36 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | -1 | 1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | 1 | 1 | -1 | -1 | 1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 1 | -1 | 1 | -i | i | 1 | ζ8 | ζ83 | ζ85 | ζ87 | -1 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ6 | 1 | -1 | 1 | i | -i | 1 | ζ83 | ζ8 | ζ87 | ζ85 | -1 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ7 | 1 | -1 | 1 | -i | i | 1 | ζ85 | ζ87 | ζ8 | ζ83 | -1 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ8 | 1 | -1 | 1 | i | -i | 1 | ζ87 | ζ85 | ζ83 | ζ8 | -1 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ9 | 4 | 4 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ10 | 4 | -4 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | -1 | -1 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ11 | 8 | 0 | -1 | 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from F9 |
| ρ12 | 8 | 0 | -1 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 3ζ5+1 | 3ζ54+1 | 3ζ52+1 | 3ζ53+1 | complex faithful |
| ρ13 | 8 | 0 | -1 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 3ζ52+1 | 3ζ53+1 | 3ζ54+1 | 3ζ5+1 | complex faithful |
| ρ14 | 8 | 0 | -1 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 3ζ53+1 | 3ζ52+1 | 3ζ5+1 | 3ζ54+1 | complex faithful |
| ρ15 | 8 | 0 | -1 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 3ζ54+1 | 3ζ5+1 | 3ζ53+1 | 3ζ52+1 | complex faithful |
►On 45 points
Generators in S45
(1 4 5 3 2)(6 35 39 24 20)(7 25 36 21 40)(8 14 26 41 37)(9 42 15 30 27)(10 31 43 28 16)(11 29 32 17 44)(12 18 22 45 33)(13 38 19 34 23)
(1 29 25)(2 11 7)(3 44 40)(4 32 36)(5 17 21)(6 37 38)(8 19 35)(9 45 16)(10 42 33)(12 31 15)(13 20 41)(14 34 39)(18 43 30)(22 28 27)(23 24 26)
(1 22 26)(2 18 14)(3 12 8)(4 45 41)(5 33 37)(6 21 42)(7 30 39)(9 20 36)(10 38 17)(11 43 34)(13 32 16)(15 35 40)(19 44 31)(23 29 28)(24 25 27)
(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)
G:=sub<Sym(45)| (1,4,5,3,2)(6,35,39,24,20)(7,25,36,21,40)(8,14,26,41,37)(9,42,15,30,27)(10,31,43,28,16)(11,29,32,17,44)(12,18,22,45,33)(13,38,19,34,23), (1,29,25)(2,11,7)(3,44,40)(4,32,36)(5,17,21)(6,37,38)(8,19,35)(9,45,16)(10,42,33)(12,31,15)(13,20,41)(14,34,39)(18,43,30)(22,28,27)(23,24,26), (1,22,26)(2,18,14)(3,12,8)(4,45,41)(5,33,37)(6,21,42)(7,30,39)(9,20,36)(10,38,17)(11,43,34)(13,32,16)(15,35,40)(19,44,31)(23,29,28)(24,25,27), (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)>;
G:=Group( (1,4,5,3,2)(6,35,39,24,20)(7,25,36,21,40)(8,14,26,41,37)(9,42,15,30,27)(10,31,43,28,16)(11,29,32,17,44)(12,18,22,45,33)(13,38,19,34,23), (1,29,25)(2,11,7)(3,44,40)(4,32,36)(5,17,21)(6,37,38)(8,19,35)(9,45,16)(10,42,33)(12,31,15)(13,20,41)(14,34,39)(18,43,30)(22,28,27)(23,24,26), (1,22,26)(2,18,14)(3,12,8)(4,45,41)(5,33,37)(6,21,42)(7,30,39)(9,20,36)(10,38,17)(11,43,34)(13,32,16)(15,35,40)(19,44,31)(23,29,28)(24,25,27), (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) );
G=PermutationGroup([[(1,4,5,3,2),(6,35,39,24,20),(7,25,36,21,40),(8,14,26,41,37),(9,42,15,30,27),(10,31,43,28,16),(11,29,32,17,44),(12,18,22,45,33),(13,38,19,34,23)], [(1,29,25),(2,11,7),(3,44,40),(4,32,36),(5,17,21),(6,37,38),(8,19,35),(9,45,16),(10,42,33),(12,31,15),(13,20,41),(14,34,39),(18,43,30),(22,28,27),(23,24,26)], [(1,22,26),(2,18,14),(3,12,8),(4,45,41),(5,33,37),(6,21,42),(7,30,39),(9,20,36),(10,38,17),(11,43,34),(13,32,16),(15,35,40),(19,44,31),(23,29,28),(24,25,27)], [(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)]])
Matrix representation of C5⋊F9 ►in GL8(𝔽241)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 240 | 240 | 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 143 | 179 | 92 | 1 | 98 | 62 | 149 | 0 |
| 24 | 111 | 106 | 129 | 196 | 64 | 133 | 91 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 143 | 179 | 92 | 1 | 98 | 62 | 149 | 0 |
| 186 | 197 | 213 | 94 | 240 | 0 | 0 | 0 |
| 147 | 92 | 103 | 119 | 0 | 240 | 0 | 0 |
| 122 | 28 | 214 | 225 | 0 | 0 | 240 | 0 |
| 82 | 93 | 213 | 229 | 175 | 70 | 182 | 1 |
| 27 | 94 | 120 | 154 | 111 | 72 | 112 | 1 |
| 221 | 29 | 228 | 207 | 86 | 130 | 23 | 91 |
| 205 | 13 | 16 | 215 | 87 | 106 | 236 | 87 |
| 194 | 76 | 125 | 174 | 174 | 43 | 87 | 205 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 24 | 111 | 106 | 129 | 196 | 64 | 133 | 91 |
| 107 | 63 | 174 | 40 | 22 | 47 | 172 | 87 |
| 148 | 192 | 166 | 28 | 66 | 171 | 59 | 240 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 240 | 240 | 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 134 | 24 | 5 | 202 | 168 | 107 | 132 | 205 |
| 24 | 111 | 106 | 129 | 196 | 64 | 133 | 91 |
| 82 | 93 | 213 | 229 | 175 | 70 | 182 | 1 |
G:=sub<GL(8,GF(241))| [0,0,0,240,0,0,143,24,1,0,0,240,0,0,179,111,0,1,0,240,0,0,92,106,0,0,1,240,0,0,1,129,0,0,0,0,0,0,98,196,0,0,0,0,1,0,62,64,0,0,0,0,0,1,149,133,0,0,0,0,0,0,0,91],[0,0,0,143,186,147,122,82,0,0,0,179,197,92,28,93,0,0,0,92,213,103,214,213,0,0,0,1,94,119,225,229,1,0,0,98,240,0,0,175,0,1,0,62,0,240,0,70,0,0,1,149,0,0,240,182,0,0,0,0,0,0,0,1],[27,221,205,194,0,24,107,148,94,29,13,76,0,111,63,192,120,228,16,125,0,106,174,166,154,207,215,174,0,129,40,28,111,86,87,174,0,196,22,66,72,130,106,43,0,64,47,171,112,23,236,87,0,133,172,59,1,91,87,205,1,91,87,240],[1,0,0,240,0,134,24,82,0,0,1,240,0,24,111,93,0,0,0,240,0,5,106,213,0,1,0,240,0,202,129,229,0,0,0,0,0,168,196,175,0,0,0,0,0,107,64,70,0,0,0,0,0,132,133,182,0,0,0,0,1,205,91,1] >;
C5⋊F9 in GAP, Magma, Sage, TeX
C_5\rtimes F_9
% in TeX
G:=Group("C5:F9"); // GroupNames label
G:=SmallGroup(360,125);
// by ID
G=gap.SmallGroup(360,125);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,3,-5,12,31,1155,681,111,1204,970,376,5189,5195]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^3,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
Export
Subgroup lattice of C5⋊F9 in TeX
Character table of C5⋊F9 in TeX