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G = C7⋊A4order 84 = 22·3·7

The semidirect product of C7 and A4 acting via A4/C22=C3

metabelian, soluble, monomial, A-group

Aliases: C7⋊A4, C22⋊(C7⋊C3), (C2×C14)⋊2C3, SmallGroup(84,11)

Series: Derived Chief Lower central Upper central

C1C2×C14 — C7⋊A4
C1C7C2×C14 — C7⋊A4
C2×C14 — C7⋊A4
C1

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 >

3C2
28C3
3C14
4C7⋊C3
7A4

Character table of C7⋊A4

 class 123A3B7A7B14A14B14C14D14E14F
 size 13282833333333
ρ1111111111111    trivial
ρ211ζ3ζ3211111111    linear of order 3
ρ311ζ32ζ311111111    linear of order 3
ρ43-10033-1-1-1-1-1-1    orthogonal lifted from A4
ρ53-100-1+-7/2-1--7/2767573ζ7472774727ζ76757374727767573    complex faithful
ρ63-100-1--7/2-1+-7/2ζ74727ζ7675737675737472776757374727    complex faithful
ρ73-100-1--7/2-1+-7/274727767573767573ζ74727ζ76757374727    complex faithful
ρ83-100-1--7/2-1+-7/274727767573ζ76757374727767573ζ74727    complex faithful
ρ93300-1+-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
ρ103-100-1+-7/2-1--7/27675737472774727767573ζ74727ζ767573    complex faithful
ρ113300-1--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
ρ123-100-1+-7/2-1--7/2ζ76757374727ζ7472776757374727767573    complex faithful

Permutation representations of C7⋊A4
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

1810
1901
100
,
15354
24306
35440
,
38322
82139
32226
,
24180
42191
4210
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

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