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G = A4×C7⋊C3order 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, C71(C3×A4), (C2×C14)⋊C32, (C22×C7⋊C3)⋊C3, C221(C3×C7⋊C3), SmallGroup(252,27)

Series: Derived Chief Lower central Upper central

C1C2×C14 — A4×C7⋊C3
C1C7C2×C14C22×C7⋊C3 — A4×C7⋊C3
C2×C14 — A4×C7⋊C3
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 >

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

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--7/2ζ3ζ763ζ753ζ73ζ3ζ743ζ723ζ7ζ32ζ7432ζ7232ζ7ζ32ζ7632ζ7532ζ73    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+-7/2ζ32ζ7432ζ7232ζ7ζ32ζ7632ζ7532ζ73ζ3ζ763ζ753ζ73ζ3ζ743ζ723ζ7    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--7/2ζ32ζ7632ζ7532ζ73ζ32ζ7432ζ7232ζ7ζ3ζ743ζ723ζ7ζ3ζ763ζ753ζ73    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+-7/2ζ3ζ743ζ723ζ7ζ3ζ763ζ753ζ73ζ32ζ7632ζ7532ζ73ζ32ζ7432ζ7232ζ7    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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