direct product, metabelian, soluble, monomial, A-group
Aliases: C2×C52⋊C3, C52⋊2C6, (C5×C10)⋊C3, SmallGroup(150,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C2×C52⋊C3 |
Generators and relations for C2×C52⋊C3
G = < a,b,c,d | a2=b5=c5=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3c3, dcd-1=b-1c >
Character table of C2×C52⋊C3
| class | 1 | 2 | 3A | 3B | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 6A | 6B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | |
| size | 1 | 1 | 25 | 25 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 25 | 25 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| ρ1 | 1 | 1 | 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 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ6 | ζ65 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ5 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ65 | ζ6 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 3 | -3 | 0 | 0 | ζ54+2ζ53 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 2ζ52+ζ5 | ζ53+2ζ5 | 2ζ54+ζ52 | 0 | 0 | -2ζ52-ζ5 | -ζ53-2ζ5 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -2ζ54-ζ52 | -ζ54-2ζ53 | complex faithful |
| ρ8 | 3 | 3 | 0 | 0 | ζ54+2ζ53 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 2ζ52+ζ5 | ζ53+2ζ5 | 2ζ54+ζ52 | 0 | 0 | 2ζ52+ζ5 | ζ53+2ζ5 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 2ζ54+ζ52 | ζ54+2ζ53 | complex lifted from C52⋊C3 |
| ρ9 | 3 | -3 | 0 | 0 | 2ζ52+ζ5 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | ζ54+2ζ53 | 2ζ54+ζ52 | ζ53+2ζ5 | 0 | 0 | -ζ54-2ζ53 | -2ζ54-ζ52 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -ζ53-2ζ5 | -2ζ52-ζ5 | complex faithful |
| ρ10 | 3 | 3 | 0 | 0 | 2ζ54+ζ52 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | ζ53+2ζ5 | ζ54+2ζ53 | 2ζ52+ζ5 | 0 | 0 | ζ53+2ζ5 | ζ54+2ζ53 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 2ζ52+ζ5 | 2ζ54+ζ52 | complex lifted from C52⋊C3 |
| ρ11 | 3 | -3 | 0 | 0 | 1+√5/2 | 2ζ54+ζ52 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ52+ζ5 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -2ζ54-ζ52 | -ζ54-2ζ53 | -ζ53-2ζ5 | -2ζ52-ζ5 | -1+√5/2 | -1-√5/2 | complex faithful |
| ρ12 | 3 | 3 | 0 | 0 | 1-√5/2 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | ζ53+2ζ5 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 0 | 0 | 1-√5/2 | 1+√5/2 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | ζ53+2ζ5 | 1+√5/2 | 1-√5/2 | complex lifted from C52⋊C3 |
| ρ13 | 3 | 3 | 0 | 0 | ζ53+2ζ5 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 2ζ54+ζ52 | 2ζ52+ζ5 | ζ54+2ζ53 | 0 | 0 | 2ζ54+ζ52 | 2ζ52+ζ5 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | ζ54+2ζ53 | ζ53+2ζ5 | complex lifted from C52⋊C3 |
| ρ14 | 3 | -3 | 0 | 0 | 2ζ54+ζ52 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | ζ53+2ζ5 | ζ54+2ζ53 | 2ζ52+ζ5 | 0 | 0 | -ζ53-2ζ5 | -ζ54-2ζ53 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -2ζ52-ζ5 | -2ζ54-ζ52 | complex faithful |
| ρ15 | 3 | 3 | 0 | 0 | 1+√5/2 | 2ζ54+ζ52 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ52+ζ5 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 0 | 0 | 1+√5/2 | 1-√5/2 | 2ζ54+ζ52 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ52+ζ5 | 1-√5/2 | 1+√5/2 | complex lifted from C52⋊C3 |
| ρ16 | 3 | 3 | 0 | 0 | 1-√5/2 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ52+ζ5 | 2ζ54+ζ52 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 0 | 0 | 1-√5/2 | 1+√5/2 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ52+ζ5 | 2ζ54+ζ52 | 1+√5/2 | 1-√5/2 | complex lifted from C52⋊C3 |
| ρ17 | 3 | -3 | 0 | 0 | 1-√5/2 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | ζ53+2ζ5 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -2ζ52-ζ5 | -2ζ54-ζ52 | -ζ54-2ζ53 | -ζ53-2ζ5 | -1-√5/2 | -1+√5/2 | complex faithful |
| ρ18 | 3 | -3 | 0 | 0 | 1+√5/2 | ζ53+2ζ5 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -ζ53-2ζ5 | -2ζ52-ζ5 | -2ζ54-ζ52 | -ζ54-2ζ53 | -1+√5/2 | -1-√5/2 | complex faithful |
| ρ19 | 3 | -3 | 0 | 0 | 1-√5/2 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ52+ζ5 | 2ζ54+ζ52 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -ζ54-2ζ53 | -ζ53-2ζ5 | -2ζ52-ζ5 | -2ζ54-ζ52 | -1-√5/2 | -1+√5/2 | complex faithful |
| ρ20 | 3 | 3 | 0 | 0 | 2ζ52+ζ5 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | ζ54+2ζ53 | 2ζ54+ζ52 | ζ53+2ζ5 | 0 | 0 | ζ54+2ζ53 | 2ζ54+ζ52 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | ζ53+2ζ5 | 2ζ52+ζ5 | complex lifted from C52⋊C3 |
| ρ21 | 3 | 3 | 0 | 0 | 1+√5/2 | ζ53+2ζ5 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 0 | 0 | 1+√5/2 | 1-√5/2 | ζ53+2ζ5 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | 1-√5/2 | 1+√5/2 | complex lifted from C52⋊C3 |
| ρ22 | 3 | -3 | 0 | 0 | ζ53+2ζ5 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 2ζ54+ζ52 | 2ζ52+ζ5 | ζ54+2ζ53 | 0 | 0 | -2ζ54-ζ52 | -2ζ52-ζ5 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -ζ54-2ζ53 | -ζ53-2ζ5 | complex faithful |
►On 30 points - transitive group 30T40
Generators in S30
(1 9)(2 8)(3 7)(4 6)(5 10)(11 30)(12 26)(13 27)(14 28)(15 29)(16 25)(17 21)(18 22)(19 23)(20 24)
(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 4 2 5 3)(6 8 10 7 9)(11 15 14 13 12)(16 18 20 17 19)(21 23 25 22 24)(26 30 29 28 27)
(1 20 26)(2 19 27)(3 18 28)(4 17 29)(5 16 30)(6 21 15)(7 22 14)(8 23 13)(9 24 12)(10 25 11)
G:=sub<Sym(30)| (1,9)(2,8)(3,7)(4,6)(5,10)(11,30)(12,26)(13,27)(14,28)(15,29)(16,25)(17,21)(18,22)(19,23)(20,24), (11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,4,2,5,3)(6,8,10,7,9)(11,15,14,13,12)(16,18,20,17,19)(21,23,25,22,24)(26,30,29,28,27), (1,20,26)(2,19,27)(3,18,28)(4,17,29)(5,16,30)(6,21,15)(7,22,14)(8,23,13)(9,24,12)(10,25,11)>;
G:=Group( (1,9)(2,8)(3,7)(4,6)(5,10)(11,30)(12,26)(13,27)(14,28)(15,29)(16,25)(17,21)(18,22)(19,23)(20,24), (11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,4,2,5,3)(6,8,10,7,9)(11,15,14,13,12)(16,18,20,17,19)(21,23,25,22,24)(26,30,29,28,27), (1,20,26)(2,19,27)(3,18,28)(4,17,29)(5,16,30)(6,21,15)(7,22,14)(8,23,13)(9,24,12)(10,25,11) );
G=PermutationGroup([[(1,9),(2,8),(3,7),(4,6),(5,10),(11,30),(12,26),(13,27),(14,28),(15,29),(16,25),(17,21),(18,22),(19,23),(20,24)], [(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30)], [(1,4,2,5,3),(6,8,10,7,9),(11,15,14,13,12),(16,18,20,17,19),(21,23,25,22,24),(26,30,29,28,27)], [(1,20,26),(2,19,27),(3,18,28),(4,17,29),(5,16,30),(6,21,15),(7,22,14),(8,23,13),(9,24,12),(10,25,11)]])
G:=TransitiveGroup(30,40);
C2×C52⋊C3 is a maximal subgroup of
C52⋊2Dic3 C52⋊2C12
Matrix representation of C2×C52⋊C3 ►in GL3(𝔽11) generated by
| 10 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 2 | 5 | 2 |
| 3 | 0 | 6 |
| 6 | 1 | 2 |
| 6 | 5 | 4 |
| 10 | 7 | 9 |
| 0 | 0 | 4 |
| 1 | 0 | 1 |
| 0 | 0 | 10 |
| 0 | 1 | 10 |
G:=sub<GL(3,GF(11))| [10,0,0,0,10,0,0,0,10],[2,3,6,5,0,1,2,6,2],[6,10,0,5,7,0,4,9,4],[1,0,0,0,0,1,1,10,10] >;
C2×C52⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_5^2\rtimes C_3
% in TeX
G:=Group("C2xC5^2:C3"); // GroupNames label
G:=SmallGroup(150,7);
// by ID
G=gap.SmallGroup(150,7);
# by ID
G:=PCGroup([4,-2,-3,-5,5,582,919]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^5=c^5=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3*c^3,d*c*d^-1=b^-1*c>;
// generators/relations
Export
Subgroup lattice of C2×C52⋊C3 in TeX
Character table of C2×C52⋊C3 in TeX