metabelian, soluble, monomial, A-group
Aliases: C7⋊A4, C22⋊(C7⋊C3), (C2×C14)⋊2C3, SmallGroup(84,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C14 — C7⋊A4 |
Generators and relations for C7⋊A4
G = < a,b,c,d | a7=b2=c2=d3=1, ab=ba, ac=ca, dad-1=a4, dbd-1=bc=cb, dcd-1=b >
Character table of C7⋊A4
| class | 1 | 2 | 3A | 3B | 7A | 7B | 14A | 14B | 14C | 14D | 14E | 14F | |
| size | 1 | 3 | 28 | 28 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 3 | -1 | 0 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ5 | 3 | -1 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -ζ76+ζ75-ζ73 | ζ74-ζ72-ζ7 | -ζ74-ζ72+ζ7 | ζ76-ζ75-ζ73 | -ζ74+ζ72-ζ7 | -ζ76-ζ75+ζ73 | complex faithful |
| ρ6 | 3 | -1 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | ζ74-ζ72-ζ7 | ζ76-ζ75-ζ73 | -ζ76+ζ75-ζ73 | -ζ74+ζ72-ζ7 | -ζ76-ζ75+ζ73 | -ζ74-ζ72+ζ7 | complex faithful |
| ρ7 | 3 | -1 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -ζ74-ζ72+ζ7 | -ζ76+ζ75-ζ73 | -ζ76-ζ75+ζ73 | ζ74-ζ72-ζ7 | ζ76-ζ75-ζ73 | -ζ74+ζ72-ζ7 | complex faithful |
| ρ8 | 3 | -1 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -ζ74+ζ72-ζ7 | -ζ76-ζ75+ζ73 | ζ76-ζ75-ζ73 | -ζ74-ζ72+ζ7 | -ζ76+ζ75-ζ73 | ζ74-ζ72-ζ7 | complex faithful |
| ρ9 | 3 | 3 | 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 |
| ρ10 | 3 | -1 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -ζ76-ζ75+ζ73 | -ζ74-ζ72+ζ7 | -ζ74+ζ72-ζ7 | -ζ76+ζ75-ζ73 | ζ74-ζ72-ζ7 | ζ76-ζ75-ζ73 | complex faithful |
| ρ11 | 3 | 3 | 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 |
| ρ12 | 3 | -1 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | ζ76-ζ75-ζ73 | -ζ74+ζ72-ζ7 | ζ74-ζ72-ζ7 | -ζ76-ζ75+ζ73 | -ζ74-ζ72+ζ7 | -ζ76+ζ75-ζ73 | complex faithful |
►On 28 points - transitive group 28T16
Generators in S28
(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)
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)
(2 3 5)(4 7 6)(8 24 21)(9 26 18)(10 28 15)(11 23 19)(12 25 16)(13 27 20)(14 22 17)
G:=sub<Sym(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), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (2,3,5)(4,7,6)(8,24,21)(9,26,18)(10,28,15)(11,23,19)(12,25,16)(13,27,20)(14,22,17)>;
G:=Group( (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), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (2,3,5)(4,7,6)(8,24,21)(9,26,18)(10,28,15)(11,23,19)(12,25,16)(13,27,20)(14,22,17) );
G=PermutationGroup([[(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)], [(1,20),(2,21),(3,15),(4,16),(5,17),(6,18),(7,19),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28)], [(1,13),(2,14),(3,8),(4,9),(5,10),(6,11),(7,12),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28)], [(2,3,5),(4,7,6),(8,24,21),(9,26,18),(10,28,15),(11,23,19),(12,25,16),(13,27,20),(14,22,17)]])
G:=TransitiveGroup(28,16);
C7⋊A4 is a maximal subgroup of
D7⋊A4 A4×C7⋊C3 C42⋊(C7⋊C3) C7⋊(C22⋊A4)
C7⋊A4 is a maximal quotient of C14.A4 C21.A4 C42⋊(C7⋊C3) C7⋊(C22⋊A4)
Matrix representation of C7⋊A4 ►in GL3(𝔽43) generated by
| 18 | 1 | 0 |
| 19 | 0 | 1 |
| 1 | 0 | 0 |
| 15 | 35 | 4 |
| 24 | 30 | 6 |
| 35 | 4 | 40 |
| 38 | 32 | 2 |
| 8 | 21 | 39 |
| 32 | 2 | 26 |
| 24 | 18 | 0 |
| 42 | 19 | 1 |
| 42 | 1 | 0 |
G:=sub<GL(3,GF(43))| [18,19,1,1,0,0,0,1,0],[15,24,35,35,30,4,4,6,40],[38,8,32,32,21,2,2,39,26],[24,42,42,18,19,1,0,1,0] >;
C7⋊A4 in GAP, Magma, Sage, TeX
C_7\rtimes A_4
% in TeX
G:=Group("C7:A4"); // GroupNames label
G:=SmallGroup(84,11);
// by ID
G=gap.SmallGroup(84,11);
# by ID
G:=PCGroup([4,-3,-2,2,-7,49,110,387]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^4,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
Export
Subgroup lattice of C7⋊A4 in TeX
Character table of C7⋊A4 in TeX