metabelian, soluble, monomial, A-group
Aliases: C52⋊2C12, (C5×C10).C6, C52⋊6C4⋊C3, C52⋊C3⋊4C4, C2.(C52⋊C6), (C2×C52⋊C3).2C2, SmallGroup(300,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C52⋊2C12 |
Generators and relations for C52⋊2C12
G = < a,b,c | a5=b5=c12=1, ab=ba, cac-1=a-1b2, cbc-1=ab2 >
Character table of C52⋊2C12
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 25 | 25 | 25 | 25 | 6 | 6 | 6 | 6 | 25 | 25 | 6 | 6 | 6 | 6 | 25 | 25 | 25 | 25 | |
| ρ1 | 1 | 1 | 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 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ7 | 1 | -1 | 1 | 1 | i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -i | i | -i | i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | -i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | i | -i | i | -i | linear of order 4 |
| ρ9 | 1 | -1 | ζ32 | ζ3 | -i | i | 1 | 1 | 1 | 1 | ζ65 | ζ6 | -1 | -1 | -1 | -1 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | linear of order 12 |
| ρ10 | 1 | -1 | ζ3 | ζ32 | -i | i | 1 | 1 | 1 | 1 | ζ6 | ζ65 | -1 | -1 | -1 | -1 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | linear of order 12 |
| ρ11 | 1 | -1 | ζ32 | ζ3 | i | -i | 1 | 1 | 1 | 1 | ζ65 | ζ6 | -1 | -1 | -1 | -1 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | linear of order 12 |
| ρ12 | 1 | -1 | ζ3 | ζ32 | i | -i | 1 | 1 | 1 | 1 | ζ6 | ζ65 | -1 | -1 | -1 | -1 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | linear of order 12 |
| ρ13 | 6 | 6 | 0 | 0 | 0 | 0 | -3-√5/2 | -3+√5/2 | 1-√5 | 1+√5 | 0 | 0 | 1-√5 | 1+√5 | -3+√5/2 | -3-√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊C6 |
| ρ14 | 6 | 6 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | -3+√5/2 | -3-√5/2 | 0 | 0 | -3+√5/2 | -3-√5/2 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊C6 |
| ρ15 | 6 | 6 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | -3-√5/2 | -3+√5/2 | 0 | 0 | -3-√5/2 | -3+√5/2 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊C6 |
| ρ16 | 6 | 6 | 0 | 0 | 0 | 0 | -3+√5/2 | -3-√5/2 | 1+√5 | 1-√5 | 0 | 0 | 1+√5 | 1-√5 | -3-√5/2 | -3+√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊C6 |
| ρ17 | 6 | -6 | 0 | 0 | 0 | 0 | -3-√5/2 | -3+√5/2 | 1-√5 | 1+√5 | 0 | 0 | -1+√5 | -1-√5 | 3-√5/2 | 3+√5/2 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ18 | 6 | -6 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | -3+√5/2 | -3-√5/2 | 0 | 0 | 3-√5/2 | 3+√5/2 | -1-√5 | -1+√5 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ19 | 6 | -6 | 0 | 0 | 0 | 0 | -3+√5/2 | -3-√5/2 | 1+√5 | 1-√5 | 0 | 0 | -1-√5 | -1+√5 | 3+√5/2 | 3-√5/2 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ20 | 6 | -6 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | -3-√5/2 | -3+√5/2 | 0 | 0 | 3+√5/2 | 3-√5/2 | -1+√5 | -1-√5 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 60 points
Generators in S60
(2 36 58 45 13)(3 25 59 46 14)(5 16 48 49 27)(6 17 37 50 28)(8 30 52 39 19)(9 31 53 40 20)(11 22 42 55 33)(12 23 43 56 34)
(1 44 35 24 57)(2 36 58 45 13)(3 46 25 14 59)(4 60 15 26 47)(5 16 48 49 27)(6 50 17 28 37)(7 38 29 18 51)(8 30 52 39 19)(9 40 31 20 53)(10 54 21 32 41)(11 22 42 55 33)(12 56 23 34 43)
(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)(49 50 51 52 53 54 55 56 57 58 59 60)
G:=sub<Sym(60)| (2,36,58,45,13)(3,25,59,46,14)(5,16,48,49,27)(6,17,37,50,28)(8,30,52,39,19)(9,31,53,40,20)(11,22,42,55,33)(12,23,43,56,34), (1,44,35,24,57)(2,36,58,45,13)(3,46,25,14,59)(4,60,15,26,47)(5,16,48,49,27)(6,50,17,28,37)(7,38,29,18,51)(8,30,52,39,19)(9,40,31,20,53)(10,54,21,32,41)(11,22,42,55,33)(12,56,23,34,43), (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)(49,50,51,52,53,54,55,56,57,58,59,60)>;
G:=Group( (2,36,58,45,13)(3,25,59,46,14)(5,16,48,49,27)(6,17,37,50,28)(8,30,52,39,19)(9,31,53,40,20)(11,22,42,55,33)(12,23,43,56,34), (1,44,35,24,57)(2,36,58,45,13)(3,46,25,14,59)(4,60,15,26,47)(5,16,48,49,27)(6,50,17,28,37)(7,38,29,18,51)(8,30,52,39,19)(9,40,31,20,53)(10,54,21,32,41)(11,22,42,55,33)(12,56,23,34,43), (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)(49,50,51,52,53,54,55,56,57,58,59,60) );
G=PermutationGroup([[(2,36,58,45,13),(3,25,59,46,14),(5,16,48,49,27),(6,17,37,50,28),(8,30,52,39,19),(9,31,53,40,20),(11,22,42,55,33),(12,23,43,56,34)], [(1,44,35,24,57),(2,36,58,45,13),(3,46,25,14,59),(4,60,15,26,47),(5,16,48,49,27),(6,50,17,28,37),(7,38,29,18,51),(8,30,52,39,19),(9,40,31,20,53),(10,54,21,32,41),(11,22,42,55,33),(12,56,23,34,43)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60)]])
Matrix representation of C52⋊2C12 ►in GL7(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 17 | 0 | 0 |
| 0 | 0 | 0 | 44 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 44 | 44 |
| 0 | 0 | 0 | 0 | 0 | 17 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 60 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 44 | 44 |
| 0 | 0 | 0 | 0 | 0 | 17 | 60 |
| 50 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 41 | 0 | 0 |
| 0 | 0 | 0 | 55 | 53 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 41 |
| 0 | 0 | 0 | 0 | 0 | 55 | 53 |
| 0 | 8 | 41 | 0 | 0 | 0 | 0 |
| 0 | 55 | 53 | 0 | 0 | 0 | 0 |
G:=sub<GL(7,GF(61))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,44,0,0,0,0,0,17,44,0,0,0,0,0,0,0,44,17,0,0,0,0,0,44,60],[1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,1,17,0,0,0,0,0,0,0,0,60,0,0,0,0,0,1,17,0,0,0,0,0,0,0,44,17,0,0,0,0,0,44,60],[50,0,0,0,0,0,0,0,0,0,0,0,8,55,0,0,0,0,0,41,53,0,8,55,0,0,0,0,0,41,53,0,0,0,0,0,0,0,8,55,0,0,0,0,0,41,53,0,0] >;
C52⋊2C12 in GAP, Magma, Sage, TeX
C_5^2\rtimes_2C_{12} % in TeX
G:=Group("C5^2:2C12"); // GroupNames label
G:=SmallGroup(300,14);
// by ID
G=gap.SmallGroup(300,14);
# by ID
G:=PCGroup([5,-2,-3,-2,-5,5,30,963,1568,6004,909]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^12=1,a*b=b*a,c*a*c^-1=a^-1*b^2,c*b*c^-1=a*b^2>;
// generators/relations
Export
Subgroup lattice of C52⋊2C12 in TeX
Character table of C52⋊2C12 in TeX