metabelian, soluble, monomial, A-group
Aliases: C32⋊F5, C5⋊(C32⋊C4), (C3×C15)⋊1C4, C3⋊D15.C2, SmallGroup(180,25)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — C32⋊F5 |
Generators and relations for C32⋊F5
G = < a,b,c,d | a3=b3=c5=d4=1, dbd-1=ab=ba, ac=ca, dad-1=a-1b, bc=cb, dcd-1=c3 >
Character table of C32⋊F5
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5 | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | |
| size | 1 | 45 | 4 | 4 | 45 | 45 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | 1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 4 | 0 | 4 | 4 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ6 | 4 | 0 | 1 | -2 | 0 | 0 | 4 | 1 | 1 | -2 | 1 | 1 | -2 | -2 | -2 | orthogonal lifted from C32⋊C4 |
| ρ7 | 4 | 0 | -2 | 1 | 0 | 0 | 4 | -2 | -2 | 1 | -2 | -2 | 1 | 1 | 1 | orthogonal lifted from C32⋊C4 |
| ρ8 | 4 | 0 | -2 | 1 | 0 | 0 | -1 | 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | -ζ32ζ54+ζ32ζ5-2ζ54-ζ5-1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52+1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | -ζ3ζ54+ζ3ζ5-2ζ54-ζ5-1 | orthogonal faithful |
| ρ9 | 4 | 0 | -2 | 1 | 0 | 0 | -1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52+1 | -ζ3ζ54+ζ3ζ5-2ζ54-ζ5-1 | 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | -ζ32ζ54+ζ32ζ5-2ζ54-ζ5-1 | orthogonal faithful |
| ρ10 | 4 | 0 | 1 | -2 | 0 | 0 | -1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | -ζ32ζ54+ζ32ζ5-2ζ54-ζ5-1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | -ζ3ζ54+ζ3ζ5-2ζ54-ζ5-1 | 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53+1 | orthogonal faithful |
| ρ11 | 4 | 0 | -2 | 1 | 0 | 0 | -1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52+1 | 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53+1 | -ζ3ζ54+ζ3ζ5-2ζ54-ζ5-1 | -ζ32ζ54+ζ32ζ5-2ζ54-ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | orthogonal faithful |
| ρ12 | 4 | 0 | 1 | -2 | 0 | 0 | -1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | -ζ3ζ54+ζ3ζ5-2ζ54-ζ5-1 | 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53+1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | -ζ32ζ54+ζ32ζ5-2ζ54-ζ5-1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | orthogonal faithful |
| ρ13 | 4 | 0 | -2 | 1 | 0 | 0 | -1 | 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52+1 | 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53+1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | -ζ32ζ54+ζ32ζ5-2ζ54-ζ5-1 | -ζ3ζ54+ζ3ζ5-2ζ54-ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | orthogonal faithful |
| ρ14 | 4 | 0 | 1 | -2 | 0 | 0 | -1 | -ζ3ζ54+ζ3ζ5-2ζ54-ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | -ζ32ζ54+ζ32ζ5-2ζ54-ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53+1 | 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52+1 | orthogonal faithful |
| ρ15 | 4 | 0 | 1 | -2 | 0 | 0 | -1 | -ζ32ζ54+ζ32ζ5-2ζ54-ζ5-1 | -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 | 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52+1 | -ζ3ζ54+ζ3ζ5-2ζ54-ζ5-1 | ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 | 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53+1 | 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 | 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 | orthogonal faithful |
►On 30 points - transitive group 30T46
Generators in S30
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(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)
(1 18)(2 20 5 16)(3 17 4 19)(6 28 11 23)(7 30 15 21)(8 27 14 24)(9 29 13 22)(10 26 12 25)
G:=sub<Sym(30)| (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (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), (1,18)(2,20,5,16)(3,17,4,19)(6,28,11,23)(7,30,15,21)(8,27,14,24)(9,29,13,22)(10,26,12,25)>;
G:=Group( (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (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), (1,18)(2,20,5,16)(3,17,4,19)(6,28,11,23)(7,30,15,21)(8,27,14,24)(9,29,13,22)(10,26,12,25) );
G=PermutationGroup([[(1,11,6),(2,12,7),(3,13,8),(4,14,9),(5,15,10),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25)], [(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(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)], [(1,18),(2,20,5,16),(3,17,4,19),(6,28,11,23),(7,30,15,21),(8,27,14,24),(9,29,13,22),(10,26,12,25)]])
G:=TransitiveGroup(30,46);
C32⋊F5 is a maximal subgroup of
C32⋊F5⋊C2
C32⋊F5 is a maximal quotient of (C3×C6).F5
Matrix representation of C32⋊F5 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 45 |
| 0 | 0 | 16 | 52 |
| 52 | 16 | 0 | 0 |
| 45 | 8 | 0 | 0 |
| 0 | 0 | 8 | 45 |
| 0 | 0 | 16 | 52 |
| 43 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 18 | 18 |
| 0 | 0 | 43 | 60 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 43 | 60 | 0 | 0 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,8,16,0,0,45,52],[52,45,0,0,16,8,0,0,0,0,8,16,0,0,45,52],[43,1,0,0,60,0,0,0,0,0,18,43,0,0,18,60],[0,0,1,43,0,0,0,60,1,0,0,0,0,1,0,0] >;
C32⋊F5 in GAP, Magma, Sage, TeX
C_3^2\rtimes F_5
% in TeX
G:=Group("C3^2:F5"); // GroupNames label
G:=SmallGroup(180,25);
// by ID
G=gap.SmallGroup(180,25);
# by ID
G:=PCGroup([5,-2,-2,-3,3,-5,10,422,67,643,248,1804,1809]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^5=d^4=1,d*b*d^-1=a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b,b*c=c*b,d*c*d^-1=c^3>;
// generators/relations
Export
Subgroup lattice of C32⋊F5 in TeX
Character table of C32⋊F5 in TeX