direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C22×C7⋊C3, C14⋊2C6, C7⋊2(C2×C6), (C2×C14)⋊3C3, SmallGroup(84,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C22×C7⋊C3 |
Generators and relations for C22×C7⋊C3
G = < a,b,c,d | a2=b2=c7=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Character table of C22×C7⋊C3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 7A | 7B | 14A | 14B | 14C | 14D | 14E | 14F | |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 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 | 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 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | ζ32 | ζ65 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 6 |
| ρ6 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | ζ3 | ζ65 | ζ32 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | linear of order 6 |
| ρ7 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 6 |
| ρ8 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | ζ3 | ζ6 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ10 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ11 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 6 |
| ρ12 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | ζ32 | ζ6 | ζ3 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | linear of order 6 |
| ρ13 | 3 | -3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 1-√-7/2 | 1+√-7/2 | -1-√-7/2 | -1+√-7/2 | 1-√-7/2 | 1+√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ14 | 3 | -3 | 3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 1-√-7/2 | -1-√-7/2 | 1+√-7/2 | 1-√-7/2 | -1+√-7/2 | 1+√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ15 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | complex lifted from C7⋊C3 |
| ρ16 | 3 | 3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | 1+√-7/2 | 1+√-7/2 | 1-√-7/2 | 1-√-7/2 | -1-√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ17 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | complex lifted from C7⋊C3 |
| ρ18 | 3 | -3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 1+√-7/2 | 1-√-7/2 | -1+√-7/2 | -1-√-7/2 | 1+√-7/2 | 1-√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ19 | 3 | 3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | 1-√-7/2 | 1-√-7/2 | 1+√-7/2 | 1+√-7/2 | -1+√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ20 | 3 | -3 | 3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 1+√-7/2 | -1+√-7/2 | 1-√-7/2 | 1+√-7/2 | -1-√-7/2 | 1-√-7/2 | complex lifted from C2×C7⋊C3 |
►On 28 points - transitive group 28T14
Generators in S28
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)
(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)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)
G:=sub<Sym(28)| (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)>;
G:=Group( (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27) );
G=PermutationGroup([[(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28)], [(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)], [(2,3,5),(4,7,6),(9,10,12),(11,14,13),(16,17,19),(18,21,20),(23,24,26),(25,28,27)]])
G:=TransitiveGroup(28,14);
C22×C7⋊C3 is a maximal subgroup of
Dic7⋊C6
Matrix representation of C22×C7⋊C3 ►in GL4(𝔽43) generated by
| 42 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 42 |
| 1 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 42 |
| 1 | 0 | 0 | 0 |
| 0 | 24 | 25 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 18 | 42 | 42 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(43))| [42,0,0,0,0,42,0,0,0,0,42,0,0,0,0,42],[1,0,0,0,0,42,0,0,0,0,42,0,0,0,0,42],[1,0,0,0,0,24,1,0,0,25,0,1,0,1,0,0],[6,0,0,0,0,1,18,0,0,0,42,1,0,0,42,0] >;
C22×C7⋊C3 in GAP, Magma, Sage, TeX
C_2^2\times C_7\rtimes C_3
% in TeX
G:=Group("C2^2xC7:C3"); // GroupNames label
G:=SmallGroup(84,9);
// by ID
G=gap.SmallGroup(84,9);
# by ID
G:=PCGroup([4,-2,-2,-3,-7,107]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^7=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations
Export
Subgroup lattice of C22×C7⋊C3 in TeX
Character table of C22×C7⋊C3 in TeX