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## G = A4×3- 1+2order 324 = 22·34

### Direct product of A4 and 3- 1+2

Aliases: A4×3- 1+2, C62.8C32, C9⋊A41C3, C91(C3×A4), (C9×A4)⋊2C3, (C2×C18)⋊C32, C3.A42C32, C32.A48C3, C3.7(C32×A4), (C2×C6).6C33, C32.8(C3×A4), (C32×A4).3C3, (C3×A4).3C32, C222(C3×3- 1+2), (C22×3- 1+2)⋊5C3, SmallGroup(324,131)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — A4×3- 1+2
 Chief series C1 — C22 — C2×C6 — C3×A4 — C32×A4 — A4×3- 1+2
 Lower central C22 — C2×C6 — A4×3- 1+2
 Upper central C1 — C3 — 3- 1+2

Generators and relations for A4×3- 1+2
G = < a,b,c,d,e | a2=b2=c3=d9=e3=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 250 in 84 conjugacy classes, 39 normal (11 characteristic)
C1, C2, C3, C3, C22, C6, C9, C9, C32, C32, A4, A4, C2×C6, C2×C6, C18, C3×C6, C3×C9, 3- 1+2, 3- 1+2, C33, C3.A4, C2×C18, C3×A4, C3×A4, C3×A4, C62, C2×3- 1+2, C3×3- 1+2, C9×A4, C9⋊A4, C32.A4, C22×3- 1+2, C32×A4, A4×3- 1+2
Quotients: C1, C3, C32, A4, 3- 1+2, C33, C3×A4, C3×3- 1+2, C32×A4, A4×3- 1+2

Smallest permutation representation of A4×3- 1+2
On 36 points
Generators in S36
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 10)(9 11)(19 32)(20 33)(21 34)(22 35)(23 36)(24 28)(25 29)(26 30)(27 31)
(1 34)(2 35)(3 36)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)
(1 4 7)(2 5 8)(3 6 9)(10 35 25)(11 36 26)(12 28 27)(13 29 19)(14 30 20)(15 31 21)(16 32 22)(17 33 23)(18 34 24)
(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 29 30 31 32 33 34 35 36)
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 25 22)(20 23 26)(29 35 32)(30 33 36)

G:=sub<Sym(36)| (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,10)(9,11)(19,32)(20,33)(21,34)(22,35)(23,36)(24,28)(25,29)(26,30)(27,31), (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27), (1,4,7)(2,5,8)(3,6,9)(10,35,25)(11,36,26)(12,28,27)(13,29,19)(14,30,20)(15,31,21)(16,32,22)(17,33,23)(18,34,24), (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)(28,29,30,31,32,33,34,35,36), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,25,22)(20,23,26)(29,35,32)(30,33,36)>;

G:=Group( (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,10)(9,11)(19,32)(20,33)(21,34)(22,35)(23,36)(24,28)(25,29)(26,30)(27,31), (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27), (1,4,7)(2,5,8)(3,6,9)(10,35,25)(11,36,26)(12,28,27)(13,29,19)(14,30,20)(15,31,21)(16,32,22)(17,33,23)(18,34,24), (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)(28,29,30,31,32,33,34,35,36), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,25,22)(20,23,26)(29,35,32)(30,33,36) );

G=PermutationGroup([[(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,10),(9,11),(19,32),(20,33),(21,34),(22,35),(23,36),(24,28),(25,29),(26,30),(27,31)], [(1,34),(2,35),(3,36),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,25),(17,26),(18,27)], [(1,4,7),(2,5,8),(3,6,9),(10,35,25),(11,36,26),(12,28,27),(13,29,19),(14,30,20),(15,31,21),(16,32,22),(17,33,23),(18,34,24)], [(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),(28,29,30,31,32,33,34,35,36)], [(2,8,5),(3,6,9),(10,16,13),(11,14,17),(19,25,22),(20,23,26),(29,35,32),(30,33,36)]])

44 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E ··· 3J 3K 3L 3M 3N 6A 6B 6C 6D 9A ··· 9F 9G ··· 9R 18A ··· 18F order 1 2 3 3 3 3 3 ··· 3 3 3 3 3 6 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 1 1 3 3 4 ··· 4 12 12 12 12 3 3 9 9 3 ··· 3 12 ··· 12 9 ··· 9

44 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 9 type + + image C1 C3 C3 C3 C3 C3 A4 3- 1+2 C3×A4 C3×A4 A4×3- 1+2 kernel A4×3- 1+2 C9×A4 C9⋊A4 C32.A4 C22×3- 1+2 C32×A4 3- 1+2 A4 C9 C32 C1 # reps 1 6 12 4 2 2 1 6 6 2 2

Matrix representation of A4×3- 1+2 in GL6(𝔽19)

 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 18 18 18 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 18 18 18
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 18 18 18
,
 11 6 0 0 0 0 7 8 1 0 0 0 7 8 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 12 11 0 0 0 0 11 0 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0,0,1,18,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,18,0,0,0,1,0,18,0,0,0,0,0,18],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,18,0,0,0,0,0,18,0,0,0,0,1,18],[11,7,7,0,0,0,6,8,8,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,12,11,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

A4×3- 1+2 in GAP, Magma, Sage, TeX

A_4\times 3_-^{1+2}
% in TeX

G:=Group("A4xES-(3,1)");
// GroupNames label

G:=SmallGroup(324,131);
// by ID

G=gap.SmallGroup(324,131);
# by ID

G:=PCGroup([6,-3,-3,-3,-3,-2,2,224,68,4864,8753]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^9=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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