A4xES-(3,1) - GroupNames

G = A₄×3_-¹⁺² order 324 = 2²·3⁴

Direct product of A₄ and 3_-¹⁺²

direct product, metabelian, soluble, monomial

Aliases: A₄×3_-¹⁺², C₆².₈C₃², C₉⋊A₄⋊₁C₃, C₉⋊₁(C₃×A₄), (C₉×A₄)⋊₂C₃, (C₂×C₁₈)⋊C₃², C₃.A₄⋊₂C₃², C₃².A₄⋊₈C₃, C₃.₇(C₃²×A₄), (C₂×C₆).₆C₃³, C₃².₈(C₃×A₄), (C₃²×A₄).₃C₃, (C₃×A₄).₃C₃², C₂²⋊₂(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

G = A4×3- 1+2 order 324 = 22·34

Direct product of A4 and 3- 1+2

G = A₄×3_-¹⁺² order 324 = 2²·3⁴

Direct product of A₄ and 3_-¹⁺²