metabelian, soluble, monomial, A-group
Aliases: C32⋊Dic5, C3⋊S3.D5, (C3×C15)⋊6C4, C5⋊2(C32⋊C4), (C5×C3⋊S3).2C2, SmallGroup(180,24)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — C32⋊Dic5 |
Generators and relations for C32⋊Dic5
G = < a,b,c,d | a3=b3=c10=1, d2=c5, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, dbd-1=a-1b-1, dcd-1=c-1 >
Character table of C32⋊Dic5
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 10A | 10B | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | |
| size | 1 | 9 | 4 | 4 | 45 | 45 | 2 | 2 | 18 | 18 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | 1 | i | -i | 1 | 1 | -1 | -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 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 2 | 2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ6 | 2 | 2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ7 | 2 | -2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ8 | 2 | -2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ9 | 4 | 0 | -2 | 1 | 0 | 0 | 4 | 4 | 0 | 0 | 1 | -2 | -2 | 1 | -2 | -2 | 1 | 1 | orthogonal lifted from C32⋊C4 |
| ρ10 | 4 | 0 | 1 | -2 | 0 | 0 | 4 | 4 | 0 | 0 | -2 | 1 | 1 | -2 | 1 | 1 | -2 | -2 | orthogonal lifted from C32⋊C4 |
| ρ11 | 4 | 0 | 1 | -2 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 1+√5/2 | -ζ53+2ζ52 | -ζ54+2ζ5 | 1-√5/2 | 2ζ54-ζ5 | 2ζ53-ζ52 | 1-√5/2 | 1+√5/2 | complex faithful |
| ρ12 | 4 | 0 | -2 | 1 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | -ζ53+2ζ52 | 1+√5/2 | 1-√5/2 | -ζ54+2ζ5 | 1-√5/2 | 1+√5/2 | 2ζ54-ζ5 | 2ζ53-ζ52 | complex faithful |
| ρ13 | 4 | 0 | 1 | -2 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 1-√5/2 | -ζ54+2ζ5 | 2ζ53-ζ52 | 1+√5/2 | -ζ53+2ζ52 | 2ζ54-ζ5 | 1+√5/2 | 1-√5/2 | complex faithful |
| ρ14 | 4 | 0 | -2 | 1 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 2ζ53-ζ52 | 1+√5/2 | 1-√5/2 | 2ζ54-ζ5 | 1-√5/2 | 1+√5/2 | -ζ54+2ζ5 | -ζ53+2ζ52 | complex faithful |
| ρ15 | 4 | 0 | 1 | -2 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 1-√5/2 | 2ζ54-ζ5 | -ζ53+2ζ52 | 1+√5/2 | 2ζ53-ζ52 | -ζ54+2ζ5 | 1+√5/2 | 1-√5/2 | complex faithful |
| ρ16 | 4 | 0 | 1 | -2 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 1+√5/2 | 2ζ53-ζ52 | 2ζ54-ζ5 | 1-√5/2 | -ζ54+2ζ5 | -ζ53+2ζ52 | 1-√5/2 | 1+√5/2 | complex faithful |
| ρ17 | 4 | 0 | -2 | 1 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | -ζ54+2ζ5 | 1-√5/2 | 1+√5/2 | 2ζ53-ζ52 | 1+√5/2 | 1-√5/2 | -ζ53+2ζ52 | 2ζ54-ζ5 | complex faithful |
| ρ18 | 4 | 0 | -2 | 1 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 2ζ54-ζ5 | 1-√5/2 | 1+√5/2 | -ζ53+2ζ52 | 1+√5/2 | 1-√5/2 | 2ζ53-ζ52 | -ζ54+2ζ5 | complex faithful |
►On 30 points - transitive group 30T48
Generators in S30
(1 21 26)(2 27 22)(3 23 28)(4 29 24)(5 25 30)(6 20 15)(7 16 11)(8 12 17)(9 18 13)(10 14 19)
(6 15 20)(7 11 16)(8 17 12)(9 13 18)(10 19 14)
(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 6)(2 10)(3 9)(4 8)(5 7)(11 25 16 30)(12 24 17 29)(13 23 18 28)(14 22 19 27)(15 21 20 26)
G:=sub<Sym(30)| (1,21,26)(2,27,22)(3,23,28)(4,29,24)(5,25,30)(6,20,15)(7,16,11)(8,12,17)(9,18,13)(10,14,19), (6,15,20)(7,11,16)(8,17,12)(9,13,18)(10,19,14), (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), (1,6)(2,10)(3,9)(4,8)(5,7)(11,25,16,30)(12,24,17,29)(13,23,18,28)(14,22,19,27)(15,21,20,26)>;
G:=Group( (1,21,26)(2,27,22)(3,23,28)(4,29,24)(5,25,30)(6,20,15)(7,16,11)(8,12,17)(9,18,13)(10,14,19), (6,15,20)(7,11,16)(8,17,12)(9,13,18)(10,19,14), (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), (1,6)(2,10)(3,9)(4,8)(5,7)(11,25,16,30)(12,24,17,29)(13,23,18,28)(14,22,19,27)(15,21,20,26) );
G=PermutationGroup([[(1,21,26),(2,27,22),(3,23,28),(4,29,24),(5,25,30),(6,20,15),(7,16,11),(8,12,17),(9,18,13),(10,14,19)], [(6,15,20),(7,11,16),(8,17,12),(9,13,18),(10,19,14)], [(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)], [(1,6),(2,10),(3,9),(4,8),(5,7),(11,25,16,30),(12,24,17,29),(13,23,18,28),(14,22,19,27),(15,21,20,26)]])
G:=TransitiveGroup(30,48);
C32⋊Dic5 is a maximal subgroup of
C5⋊F9 D5×C32⋊C4 S32⋊D5 C32⋊Dic10
C32⋊Dic5 is a maximal quotient of (C3×C15)⋊9C8
Matrix representation of C32⋊Dic5 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 60 |
| 0 | 1 | 0 | 0 |
| 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 60 |
| 34 | 0 | 0 | 0 |
| 27 | 27 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 52 | 52 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,60],[0,60,0,0,1,60,0,0,0,0,0,60,0,0,1,60],[34,27,0,0,0,27,0,0,0,0,9,52,0,0,0,52],[0,0,1,60,0,0,0,60,1,0,0,0,0,1,0,0] >;
C32⋊Dic5 in GAP, Magma, Sage, TeX
C_3^2\rtimes {\rm Dic}_5 % in TeX
G:=Group("C3^2:Dic5"); // GroupNames label
G:=SmallGroup(180,24);
// by ID
G=gap.SmallGroup(180,24);
# by ID
G:=PCGroup([5,-2,-2,-3,3,-5,10,302,67,323,248,3604]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^10=1,d^2=c^5,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of C32⋊Dic5 in TeX
Character table of C32⋊Dic5 in TeX