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G = A4×C7⋊C3  order 252 = 22·32·7

Direct product of A4 and C7⋊C3

direct product, metabelian, soluble, monomial, A-group

Aliases: A4×C7⋊C3, C7⋊A4⋊C3, (C7×A4)⋊C3, C7⋊1(C3×A4), (C2×C14)⋊C32, (C22×C7⋊C3)⋊C3, C22⋊1(C3×C7⋊C3), SmallGroup(252,27)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — A4×C7⋊C3
Chief series
C1 — C7 — C2×C14 — C22×C7⋊C3 — A4×C7⋊C3
Lower central
C2×C14 — A4×C7⋊C3
Upper central
C1

Generators and relations for A4×C7⋊C3
 G = < a,b,c,d,e | a2=b2=c3=d7=e3=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

C1
3C2
4C3
7C3
28C3
28C3
C7
C22
21C6
28C32
3C14
C7⋊C3
4C7⋊C3
4C7⋊C3
4C21
A4
7C2×C6
7A4
7A4
C2×C14
3C2×C7⋊C3
4C3×C7⋊C3
7C3×A4
C7×A4
C22×C7⋊C3
C7⋊A4
C7⋊A4
A4×C7⋊C3

Character table of A4×C7⋊C3

 class 123A3B3C3D3E3F3G3H6A6B7A7B14A14B21A21B21C21D
 size 134477282828282121339912121212
ρ111111111111111111111    trivial
ρ211ζ32ζ3ζ3ζ32ζ3ζ3211ζ32ζ31111ζ32ζ32ζ3ζ3    linear of order 3
ρ311ζ32ζ311ζ32ζ3ζ3ζ32111111ζ32ζ32ζ3ζ3    linear of order 3
ρ41111ζ3ζ32ζ32ζ3ζ32ζ3ζ32ζ311111111    linear of order 3
ρ511ζ32ζ3ζ32ζ311ζ32ζ3ζ3ζ321111ζ32ζ32ζ3ζ3    linear of order 3
ρ611ζ3ζ32ζ3ζ3211ζ3ζ32ζ32ζ31111ζ3ζ3ζ32ζ32    linear of order 3
ρ71111ζ32ζ3ζ3ζ32ζ3ζ32ζ3ζ3211111111    linear of order 3
ρ811ζ3ζ32ζ32ζ3ζ32ζ311ζ3ζ321111ζ3ζ3ζ32ζ32    linear of order 3
ρ911ζ3ζ3211ζ3ζ32ζ32ζ3111111ζ3ζ3ζ32ζ32    linear of order 3
ρ103-100330000-1-133-1-10000    orthogonal lifted from A4
ρ113-100-3-3√-3/2-3+3√-3/20000ζ65ζ633-1-10000    complex lifted from C3×A4
ρ123-100-3+3√-3/2-3-3√-3/20000ζ6ζ6533-1-10000    complex lifted from C3×A4
ρ13333300000000-1+√-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
ρ14333300000000-1-√-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
ρ1533-3+3√-3/2-3-3√-3/200000000-1-√-7/2-1+√-7/2-1+√-7/2-1-√-7/2ζ3ζ76+ζ3ζ75+ζ3ζ73ζ3ζ74+ζ3ζ72+ζ3ζ7ζ32ζ74+ζ32ζ72+ζ32ζ7ζ32ζ76+ζ32ζ75+ζ32ζ73    complex lifted from C3×C7⋊C3
ρ1633-3-3√-3/2-3+3√-3/200000000-1+√-7/2-1-√-7/2-1-√-7/2-1+√-7/2ζ32ζ74+ζ32ζ72+ζ32ζ7ζ32ζ76+ζ32ζ75+ζ32ζ73ζ3ζ76+ζ3ζ75+ζ3ζ73ζ3ζ74+ζ3ζ72+ζ3ζ7    complex lifted from C3×C7⋊C3
ρ1733-3-3√-3/2-3+3√-3/200000000-1-√-7/2-1+√-7/2-1+√-7/2-1-√-7/2ζ32ζ76+ζ32ζ75+ζ32ζ73ζ32ζ74+ζ32ζ72+ζ32ζ7ζ3ζ74+ζ3ζ72+ζ3ζ7ζ3ζ76+ζ3ζ75+ζ3ζ73    complex lifted from C3×C7⋊C3
ρ1833-3+3√-3/2-3-3√-3/200000000-1+√-7/2-1-√-7/2-1-√-7/2-1+√-7/2ζ3ζ74+ζ3ζ72+ζ3ζ7ζ3ζ76+ζ3ζ75+ζ3ζ73ζ32ζ76+ζ32ζ75+ζ32ζ73ζ32ζ74+ζ32ζ72+ζ32ζ7    complex lifted from C3×C7⋊C3
ρ199-30000000000-3-3√-7/2-3+3√-7/21-√-7/21+√-7/20000    complex faithful
ρ209-30000000000-3+3√-7/2-3-3√-7/21+√-7/21-√-7/20000    complex faithful

Permutation representations of A4×C7⋊C3
►On 28 points - transitive group 28T40
Generators in S28
(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 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)

(8 15 22)(9 16 23)(10 17 24)(11 18 25)(12 19 26)(13 20 27)(14 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)| (1,8)(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), (1,15)(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), (8,15,22)(9,16,23)(10,17,24)(11,18,25)(12,19,26)(13,20,27)(14,21,28), (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( (1,8)(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), (1,15)(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), (8,15,22)(9,16,23)(10,17,24)(11,18,25)(12,19,26)(13,20,27)(14,21,28), (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([[(1,8),(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)], [(1,15),(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)], [(8,15,22),(9,16,23),(10,17,24),(11,18,25),(12,19,26),(13,20,27),(14,21,28)], [(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(28,40);
 

Matrix representation of A4×C7⋊C3 ►in GL6(𝔽43)

100000
010000
001000
0004200
0004201
0004210
,
100000
010000
001000
0000421
0000420
0001420
,
3600000
0360000
0036000
0000036
0003600
0000360
,
42241000
0241000
42251000
000100
000010
000001
,
403639000
3600000
36363000
0003600
0000360
0000036

G:=sub<GL(6,GF(43))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,42,42,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,42,42,42,0,0,0,1,0,0],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,36,0,0],[42,0,42,0,0,0,24,24,25,0,0,0,1,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,36,36,0,0,0,36,0,36,0,0,0,39,0,3,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36] >;
 

A4×C7⋊C3 in GAP, Magma, Sage, TeX 

A_4\times C_7\rtimes C_3
 
% in TeX
 
G:=Group("A4xC7:C3");
 
// GroupNames label
 
G:=SmallGroup(252,27);
 
// by ID
 
G=gap.SmallGroup(252,27);
 
# by ID
 
G:=PCGroup([5,-3,-3,-2,2,-7,142,68,1804]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^7=e^3=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^4>;
 
// generators/relations
 

Export

Subgroup lattice of A4×C7⋊C3 in TeX
Character table of A4×C7⋊C3 in TeX

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