D9:A4 - GroupNames

G = D₉⋊A₄ order 216 = 2³·3³

The semidirect product of D₉ and A₄ acting via A₄/C₂²=C₃

Aliases: D₉⋊A₄, C₉⋊A₄⋊C₂, C₉⋊(C₂×A₄), (C₃×A₄).S₃, (C₂×C₁₈)⋊₁C₆, C₃.₁(S₃×A₄), C₂²⋊₂(C₉⋊C₆), (C₂²×D₉)⋊₁C₃, (C₂×C₆).₅(C₃×S₃), SmallGroup(216,96)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₈ — D₉⋊A₄

Chief series

C₁ — C₃ — C₉ — C₂×C₁₈ — C₉⋊A₄ — D₉⋊A₄

Lower central

C₂×C₁₈ — D₉⋊A₄

Upper central

C₁

Generators and relations for D₉⋊A₄
G = < a,b,c,d,e | a⁹=b²=c²=d²=e³=1, bab=a^-1, ac=ca, ad=da, eae^-1=a⁴, bc=cb, bd=db, ebe^-1=a³b, ece^-1=cd=dc, ede^-1=c >

C₁

₃C₂

₉C₂

₂₇C₂

C₃

₁₂C₃

C₂²

₂₇C₂²

₃S₃

₃C₆

₉S₃

₃₆C₆

C₉

₄C₃²

₈C₉

₉C₂³

C₂×C₆

₃A₄

₉D₆

D₉

₃D₉

₃C₁₈

₁₂C₃×S₃

₄3_-¹⁺²

₃C₂²×S₃

₉C₂×A₄

C₃×A₄

C₂×C₁₈

₂C₃.A₄

₃D₁₈

₄C₉⋊C₆

C₂²×D₉

₃S₃×A₄

C₉⋊A₄

D₉⋊A₄

Character table of D₉⋊A₄

class	1	2A	2B	2C	3A	3B	3C	6A	6B	6C	9A	9B	9C	18A	18B	18C
size	1	3	9	27	2	12	12	6	36	36	6	24	24	6	6	6
ρ₁	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	linear of order 2
ρ₃	1	1	1	1	1	ζ₃	ζ₃²	1	ζ₃²	ζ₃	1	ζ₃²	ζ₃	1	1	1	linear of order 3
ρ₄	1	1	1	1	1	ζ₃²	ζ₃	1	ζ₃	ζ₃²	1	ζ₃	ζ₃²	1	1	1	linear of order 3
ρ₅	1	1	-1	-1	1	ζ₃²	ζ₃	1	ζ₆⁵	ζ₆	1	ζ₃	ζ₃²	1	1	1	linear of order 6
ρ₆	1	1	-1	-1	1	ζ₃	ζ₃²	1	ζ₆	ζ₆⁵	1	ζ₃²	ζ₃	1	1	1	linear of order 6
ρ₇	2	2	0	0	2	2	2	2	0	0	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₈	2	2	0	0	2	-1+√-3	-1-√-3	2	0	0	-1	ζ₆	ζ₆⁵	-1	-1	-1	complex lifted from C₃×S₃
ρ₉	2	2	0	0	2	-1-√-3	-1+√-3	2	0	0	-1	ζ₆⁵	ζ₆	-1	-1	-1	complex lifted from C₃×S₃
ρ₁₀	3	-1	-3	1	3	0	0	-1	0	0	3	0	0	-1	-1	-1	orthogonal lifted from C₂×A₄
ρ₁₁	3	-1	3	-1	3	0	0	-1	0	0	3	0	0	-1	-1	-1	orthogonal lifted from A₄
ρ₁₂	6	6	0	0	-3	0	0	-3	0	0	0	0	0	0	0	0	orthogonal lifted from C₉⋊C₆
ρ₁₃	6	-2	0	0	6	0	0	-2	0	0	-3	0	0	1	1	1	orthogonal lifted from S₃×A₄
ρ₁₄	6	-2	0	0	-3	0	0	1	0	0	0	0	0	2ζ₉⁷+2ζ₉²	2ζ₉⁵+2ζ₉⁴	2ζ₉⁸+2ζ₉	orthogonal faithful
ρ₁₅	6	-2	0	0	-3	0	0	1	0	0	0	0	0	2ζ₉⁸+2ζ₉	2ζ₉⁷+2ζ₉²	2ζ₉⁵+2ζ₉⁴	orthogonal faithful
ρ₁₆	6	-2	0	0	-3	0	0	1	0	0	0	0	0	2ζ₉⁵+2ζ₉⁴	2ζ₉⁸+2ζ₉	2ζ₉⁷+2ζ₉²	orthogonal faithful

Smallest permutation representation of D₉⋊A₄
►On 36 points

Generators in S₃₆

(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 14)(11 13)(15 18)(16 17)(19 23)(20 22)(24 27)(25 26)(28 32)(29 31)(33 36)(34 35)

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

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

(2 8 5)(3 6 9)(10 22 34)(11 20 29)(12 27 33)(13 25 28)(14 23 32)(15 21 36)(16 19 31)(17 26 35)(18 24 30)

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

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

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

D₉⋊A₄ is a maximal quotient of Dic₉.A₄ D₁₈.A₄ Dic₉⋊A₄

Matrix representation of D₉⋊A₄ ►in GL₆(𝔽₁₉)

0	0	18	18	0	0
0	0	1	0	0	0
0	0	0	0	18	18
0	0	0	0	1	0
0	1	0	0	0	0
18	18	0	0	0	0

0	0	18	18	0	0
0	0	0	1	0	0
18	18	0	0	0	0
0	1	0	0	0	0
0	0	0	0	1	0
0	0	0	0	18	18

6	0	3	14	14	3
0	6	5	8	16	11
8	5	6	0	3	14
14	3	0	6	5	8
11	16	8	5	6	0
3	14	14	3	0	6

6	0	5	8	8	5
0	6	11	16	14	3
16	11	6	0	5	8
8	5	0	6	11	16
3	14	16	11	6	0
5	8	8	5	0	6

1	0	0	0	0	0
0	1	0	0	0	0
0	0	18	18	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	18	18

G:=sub<GL(6,GF(19))| [0,0,0,0,0,18,0,0,0,0,1,18,18,1,0,0,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0],[0,0,18,0,0,0,0,0,18,1,0,0,18,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,18],[6,0,8,14,11,3,0,6,5,3,16,14,3,5,6,0,8,14,14,8,0,6,5,3,14,16,3,5,6,0,3,11,14,8,0,6],[6,0,16,8,3,5,0,6,11,5,14,8,5,11,6,0,16,8,8,16,0,6,11,5,8,14,5,11,6,0,5,3,8,16,0,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,1,18] >;

D₉⋊A₄ in GAP, Magma, Sage, TeX

D_9\rtimes A_4

% in TeX

G:=Group("D9:A4");

// GroupNames label

G:=SmallGroup(216,96);

// by ID

G=gap.SmallGroup(216,96);

# by ID

G:=PCGroup([6,-2,-3,-2,2,-3,-3,170,81,3604,1450,208,5189]);

// Polycyclic

G:=Group<a,b,c,d,e|a^9=b^2=c^2=d^2=e^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^4,b*c=c*b,b*d=d*b,e*b*e^-1=a^3*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;

// generators/relations

Export

Subgroup lattice of D₉⋊A₄ in TeX
Character table of D₉⋊A₄ in TeX

G = D9⋊A4 order 216 = 23·33

The semidirect product of D9 and A4 acting via A4/C22=C3

G = D₉⋊A₄ order 216 = 2³·3³

The semidirect product of D₉ and A₄ acting via A₄/C₂²=C₃