Copied to
clipboard

## G = A4×D9order 216 = 23·33

### Direct product of A4 and D9

Aliases: A4×D9, C93(C2×A4), (C9×A4)⋊3C2, (C2×C18)⋊2C6, C3.2(S3×A4), (C3×A4).3S3, C222(C3×D9), (C22×D9)⋊2C3, (C2×C6).6(C3×S3), SmallGroup(216,97)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — A4×D9
 Chief series C1 — C3 — C9 — C2×C18 — C9×A4 — A4×D9
 Lower central C2×C18 — A4×D9
 Upper central C1

Generators and relations for A4×D9
G = < a,b,c,d,e | a2=b2=c3=d9=e2=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

3C2
9C2
27C2
4C3
8C3
27C22
27C22
3C6
3S3
9S3
36C6
4C32
8C9
9C23
2A4
9D6
9D6
3D9
3C18
12C3×S3
3D18
3D18

Character table of A4×D9

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 6A 6B 6C 9A 9B 9C 9D 9E 9F 9G 9H 9I 18A 18B 18C size 1 3 9 27 2 4 4 8 8 6 36 36 2 2 2 8 8 8 8 8 8 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ3 ζ32 1 1 1 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 1 1 1 linear of order 3 ρ4 1 1 -1 -1 1 ζ32 ζ3 ζ32 ζ3 1 ζ6 ζ65 1 1 1 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 1 1 1 linear of order 6 ρ5 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ32 ζ3 1 1 1 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 1 1 1 linear of order 3 ρ6 1 1 -1 -1 1 ζ3 ζ32 ζ3 ζ32 1 ζ65 ζ6 1 1 1 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 1 1 1 linear of order 6 ρ7 2 2 0 0 2 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 0 0 -1 2 2 -1 -1 -1 0 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ9 2 2 0 0 -1 2 2 -1 -1 -1 0 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ10 2 2 0 0 -1 2 2 -1 -1 -1 0 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ11 2 2 0 0 2 -1-√-3 -1+√-3 -1-√-3 -1+√-3 2 0 0 -1 -1 -1 ζ6 ζ6 ζ65 ζ6 ζ65 ζ65 -1 -1 -1 complex lifted from C3×S3 ρ12 2 2 0 0 2 -1+√-3 -1-√-3 -1+√-3 -1-√-3 2 0 0 -1 -1 -1 ζ65 ζ65 ζ6 ζ65 ζ6 ζ6 -1 -1 -1 complex lifted from C3×S3 ρ13 2 2 0 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 -1 0 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ95 ζ92+ζ9 ζ98+ζ97 ζ98+ζ94 ζ95+ζ9 ζ94+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 complex lifted from C3×D9 ρ14 2 2 0 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 -1 0 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ97 ζ95+ζ9 ζ98+ζ94 ζ94+ζ92 ζ97+ζ95 ζ92+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 complex lifted from C3×D9 ρ15 2 2 0 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 -1 0 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ9 ζ94+ζ92 ζ97+ζ95 ζ98+ζ97 ζ92+ζ9 ζ98+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 complex lifted from C3×D9 ρ16 2 2 0 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 -1 0 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ94 ζ97+ζ95 ζ94+ζ92 ζ92+ζ9 ζ98+ζ97 ζ95+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 complex lifted from C3×D9 ρ17 2 2 0 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 -1 0 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ92+ζ9 ζ98+ζ94 ζ95+ζ9 ζ97+ζ95 ζ94+ζ92 ζ98+ζ97 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 complex lifted from C3×D9 ρ18 2 2 0 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 -1 0 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ94+ζ92 ζ98+ζ97 ζ92+ζ9 ζ95+ζ9 ζ98+ζ94 ζ97+ζ95 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 complex lifted from C3×D9 ρ19 3 -1 -3 1 3 0 0 0 0 -1 0 0 3 3 3 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from C2×A4 ρ20 3 -1 3 -1 3 0 0 0 0 -1 0 0 3 3 3 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from A4 ρ21 6 -2 0 0 6 0 0 0 0 -2 0 0 -3 -3 -3 0 0 0 0 0 0 1 1 1 orthogonal lifted from S3×A4 ρ22 6 -2 0 0 -3 0 0 0 0 1 0 0 3ζ97+3ζ92 3ζ95+3ζ94 3ζ98+3ζ9 0 0 0 0 0 0 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 orthogonal faithful ρ23 6 -2 0 0 -3 0 0 0 0 1 0 0 3ζ95+3ζ94 3ζ98+3ζ9 3ζ97+3ζ92 0 0 0 0 0 0 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 orthogonal faithful ρ24 6 -2 0 0 -3 0 0 0 0 1 0 0 3ζ98+3ζ9 3ζ97+3ζ92 3ζ95+3ζ94 0 0 0 0 0 0 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 orthogonal faithful

Smallest permutation representation of A4×D9
On 36 points
Generators in S36
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)
(10 19 28)(11 20 29)(12 21 30)(13 22 31)(14 23 32)(15 24 33)(16 25 34)(17 26 35)(18 27 36)
(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)
(1 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)(28 35)(29 34)(30 33)(31 32)

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

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

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

A4×D9 is a maximal quotient of   Dic9.2A4

Matrix representation of A4×D9 in GL5(𝔽19)

 1 0 0 0 0 0 1 0 0 0 0 0 18 0 0 0 0 18 0 1 0 0 18 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 18 1 0 0 0 18 0 0 0 1 18 0
,
 7 0 0 0 0 0 7 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 12 17 0 0 0 2 14 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 2 14 0 0 0 12 17 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,18,18,18,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,18,18,18,0,0,1,0,0],[7,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[12,2,0,0,0,17,14,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[2,12,0,0,0,14,17,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4×D9 in GAP, Magma, Sage, TeX

A_4\times D_9
% in TeX

G:=Group("A4xD9");
// GroupNames label

G:=SmallGroup(216,97);
// by ID

G=gap.SmallGroup(216,97);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-3,-3,170,81,3604,208,5189]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^9=e^2=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=d^-1>;
// generators/relations

Export

׿
×
𝔽