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## G = A4×C7⋊C3order 252 = 22·32·7

### Direct product of A4 and C7⋊C3

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

 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 >

Character table of A4×C7⋊C3

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 7A 7B 14A 14B 21A 21B 21C 21D size 1 3 4 4 7 7 28 28 28 28 21 21 3 3 9 9 12 12 12 12 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 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ3 1 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ4 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 1 1 linear of order 3 ρ5 1 1 ζ32 ζ3 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ6 1 1 ζ3 ζ32 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ7 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 1 1 linear of order 3 ρ8 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ9 1 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ10 3 -1 0 0 3 3 0 0 0 0 -1 -1 3 3 -1 -1 0 0 0 0 orthogonal lifted from A4 ρ11 3 -1 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ65 ζ6 3 3 -1 -1 0 0 0 0 complex lifted from C3×A4 ρ12 3 -1 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ6 ζ65 3 3 -1 -1 0 0 0 0 complex lifted from C3×A4 ρ13 3 3 3 3 0 0 0 0 0 0 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 ρ14 3 3 3 3 0 0 0 0 0 0 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 ρ15 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 complex lifted from C3×C7⋊C3 ρ16 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 complex lifted from C3×C7⋊C3 ρ17 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 complex lifted from C3×C7⋊C3 ρ18 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 complex lifted from C3×C7⋊C3 ρ19 9 -3 0 0 0 0 0 0 0 0 0 0 -3-3√-7/2 -3+3√-7/2 1-√-7/2 1+√-7/2 0 0 0 0 complex faithful ρ20 9 -3 0 0 0 0 0 0 0 0 0 0 -3+3√-7/2 -3-3√-7/2 1+√-7/2 1-√-7/2 0 0 0 0 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)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 42 0 0 0 0 0 42 0 1 0 0 0 42 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 42 1 0 0 0 0 42 0 0 0 0 1 42 0
,
 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 36 0 0 0 0 0 0 36 0
,
 42 24 1 0 0 0 0 24 1 0 0 0 42 25 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 36 39 0 0 0 36 0 0 0 0 0 36 36 3 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36

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

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