direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C4×C7⋊C3, C28⋊C3, C7⋊2C12, C14.2C6, C2.(C2×C7⋊C3), (C2×C7⋊C3).2C2, SmallGroup(84,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C4×C7⋊C3 |
Generators and relations for C4×C7⋊C3
G = < a,b,c | a4=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Character table of C4×C7⋊C3
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 7A | 7B | 12A | 12B | 12C | 12D | 14A | 14B | 28A | 28B | 28C | 28D | |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 7 | 7 | 3 | 3 | 7 | 7 | 7 | 7 | 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 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | ζ32 | ζ3 | 1 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | ζ3 | ζ32 | 1 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 6 |
| ρ7 | 1 | -1 | 1 | 1 | i | -i | -1 | -1 | 1 | 1 | -i | -i | i | i | -1 | -1 | i | -i | i | -i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | 1 | 1 | i | i | -i | -i | -1 | -1 | -i | i | -i | i | linear of order 4 |
| ρ9 | 1 | -1 | ζ32 | ζ3 | -i | i | ζ6 | ζ65 | 1 | 1 | ζ4ζ3 | ζ4ζ32 | ζ43ζ3 | ζ43ζ32 | -1 | -1 | -i | i | -i | i | linear of order 12 |
| ρ10 | 1 | -1 | ζ3 | ζ32 | -i | i | ζ65 | ζ6 | 1 | 1 | ζ4ζ32 | ζ4ζ3 | ζ43ζ32 | ζ43ζ3 | -1 | -1 | -i | i | -i | i | linear of order 12 |
| ρ11 | 1 | -1 | ζ3 | ζ32 | i | -i | ζ65 | ζ6 | 1 | 1 | ζ43ζ32 | ζ43ζ3 | ζ4ζ32 | ζ4ζ3 | -1 | -1 | i | -i | i | -i | linear of order 12 |
| ρ12 | 1 | -1 | ζ32 | ζ3 | i | -i | ζ6 | ζ65 | 1 | 1 | ζ43ζ3 | ζ43ζ32 | ζ4ζ3 | ζ4ζ32 | -1 | -1 | i | -i | i | -i | linear of order 12 |
| ρ13 | 3 | 3 | 0 | 0 | 3 | 3 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | complex lifted from C7⋊C3 |
| ρ14 | 3 | 3 | 0 | 0 | 3 | 3 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | complex lifted from C7⋊C3 |
| ρ15 | 3 | 3 | 0 | 0 | -3 | -3 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 1-√-7/2 | 1-√-7/2 | 1+√-7/2 | 1+√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ16 | 3 | 3 | 0 | 0 | -3 | -3 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 0 | 0 | -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 | 0 | 0 | 3i | -3i | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 0 | 0 | 1+√-7/2 | 1-√-7/2 | ζ4ζ74+ζ4ζ72+ζ4ζ7 | ζ43ζ74+ζ43ζ72+ζ43ζ7 | ζ4ζ76+ζ4ζ75+ζ4ζ73 | ζ43ζ76+ζ43ζ75+ζ43ζ73 | complex faithful |
| ρ18 | 3 | -3 | 0 | 0 | -3i | 3i | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 0 | 0 | 1+√-7/2 | 1-√-7/2 | ζ43ζ74+ζ43ζ72+ζ43ζ7 | ζ4ζ74+ζ4ζ72+ζ4ζ7 | ζ43ζ76+ζ43ζ75+ζ43ζ73 | ζ4ζ76+ζ4ζ75+ζ4ζ73 | complex faithful |
| ρ19 | 3 | -3 | 0 | 0 | -3i | 3i | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 0 | 0 | 1-√-7/2 | 1+√-7/2 | ζ43ζ76+ζ43ζ75+ζ43ζ73 | ζ4ζ76+ζ4ζ75+ζ4ζ73 | ζ43ζ74+ζ43ζ72+ζ43ζ7 | ζ4ζ74+ζ4ζ72+ζ4ζ7 | complex faithful |
| ρ20 | 3 | -3 | 0 | 0 | 3i | -3i | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 0 | 0 | 1-√-7/2 | 1+√-7/2 | ζ4ζ76+ζ4ζ75+ζ4ζ73 | ζ43ζ76+ζ43ζ75+ζ43ζ73 | ζ4ζ74+ζ4ζ72+ζ4ζ7 | ζ43ζ74+ζ43ζ72+ζ43ζ7 | complex faithful |
►On 28 points - transitive group 28T13
Generators in S28
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)
(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)| (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21), (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( (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21), (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([[(1,22,8,15),(2,23,9,16),(3,24,10,17),(4,25,11,18),(5,26,12,19),(6,27,13,20),(7,28,14,21)], [(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(28,13);
C4×C7⋊C3 is a maximal subgroup of
C7⋊C24 C4.F7 C4⋊F7 C28.A4
Matrix representation of C4×C7⋊C3 ►in GL4(𝔽337) generated by
| 189 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 336 | 124 | 1 |
| 0 | 0 | 124 | 1 |
| 0 | 336 | 125 | 1 |
| 208 | 0 | 0 | 0 |
| 0 | 125 | 1 | 213 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 212 |
G:=sub<GL(4,GF(337))| [189,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,336,0,336,0,124,124,125,0,1,1,1],[208,0,0,0,0,125,1,1,0,1,0,1,0,213,0,212] >;
C4×C7⋊C3 in GAP, Magma, Sage, TeX
C_4\times C_7\rtimes C_3
% in TeX
G:=Group("C4xC7:C3"); // GroupNames label
G:=SmallGroup(84,2);
// by ID
G=gap.SmallGroup(84,2);
# by ID
G:=PCGroup([4,-2,-3,-2,-7,24,199]);
// Polycyclic
G:=Group<a,b,c|a^4=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
Export
Subgroup lattice of C4×C7⋊C3 in TeX
Character table of C4×C7⋊C3 in TeX