He3.D6 - GroupNames

G = He₃.D₆ order 324 = 2²·3⁴

1^st non-split extension by He₃ of D₆ acting faithfully

Aliases: He₃.₁D₆, C₉⋊S₃⋊₂S₃, (C₃×C₉)⋊₂D₆, C₃².₂S₃², He₃⋊C₂.S₃, He₃.C₃⋊C₂², He₃.S₃⋊C₂, He₃.C₆⋊C₂, He₃.₃S₃⋊C₂, C₃.₃(C₃²⋊D₆), SmallGroup(324,40)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — He₃.C₃ — He₃.D₆

Chief series

C₁ — C₃ — C₃² — He₃ — He₃.C₃ — He₃.C₆ — He₃.D₆

Lower central

He₃.C₃ — He₃.D₆

Upper central

C₁

Generators and relations for He₃.D₆
G = < a,b,c,d,e | a³=b³=c³=e²=1, d⁶=ebe=b^-1, ab=ba, cac^-1=eae=ab^-1, dad^-1=a^-1b, bc=cb, bd=db, dcd^-1=ece=a^-1c^-1, ede=bd⁵ >

C₁

₉C₂

₂₇C₂

C₃

₃C₃

₉C₃

₈₁C₂²

₃S₃

₉S₃

₉C₆

₉S₃

₂₇C₆

₂₇S₃

₂₇C₆

₂₇S₃

C₃²

₃C₃²

₃C₉

₆C₉

₂₇D₆

₃C₃⋊S₃

₃D₉

₃C₃⋊S₃

₃C₃×S₃

₆D₉

₉D₉

₉C₃×S₃

₉C₁₈

₉C₃×S₃

He₃

C₃×C₉

₂3_-¹⁺²

₉S₃²

₉D₁₈

C₉⋊S₃

He₃⋊C₂

₃C₃²⋊C₆

₃C₃×D₉

₃C₃²⋊C₆

₃S₃×C₉

₆C₉⋊C₆

He₃.C₃

₃C₃²⋊D₆

₃S₃×D₉

He₃.₃S₃

He₃.S₃

He₃.C₆

He₃.D₆

Character table of He₃.D₆

class	1	2A	2B	2C	3A	3B	3C	6A	6B	6C	9A	9B	9C	9D	18A	18B	18C
size	1	9	27	27	2	6	18	18	54	54	6	6	6	36	18	18	18
ρ₁	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	linear of order 2
ρ₃	1	-1	1	-1	1	1	1	-1	-1	1	1	1	1	1	-1	-1	-1	linear of order 2
ρ₄	1	1	-1	-1	1	1	1	1	-1	-1	1	1	1	1	1	1	1	linear of order 2
ρ₅	2	0	2	0	2	2	-1	0	0	-1	2	2	2	-1	0	0	0	orthogonal lifted from S₃
ρ₆	2	2	0	0	2	2	2	2	0	0	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₇	2	0	-2	0	2	2	-1	0	0	1	2	2	2	-1	0	0	0	orthogonal lifted from D₆
ρ₈	2	-2	0	0	2	2	2	-2	0	0	-1	-1	-1	-1	1	1	1	orthogonal lifted from D₆
ρ₉	4	0	0	0	4	4	-2	0	0	0	-2	-2	-2	1	0	0	0	orthogonal lifted from S₃²
ρ₁₀	6	0	0	2	6	-3	0	0	-1	0	0	0	0	0	0	0	0	orthogonal lifted from C₃²⋊D₆
ρ₁₁	6	0	0	-2	6	-3	0	0	1	0	0	0	0	0	0	0	0	orthogonal lifted from C₃²⋊D₆
ρ₁₂	6	-2	0	0	-3	0	0	1	0	0	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	0	-ζ₉⁸-ζ₉	-ζ₉⁷-ζ₉²	-ζ₉⁵-ζ₉⁴	orthogonal faithful
ρ₁₃	6	2	0	0	-3	0	0	-1	0	0	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	0	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	orthogonal faithful
ρ₁₄	6	2	0	0	-3	0	0	-1	0	0	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	0	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	orthogonal faithful
ρ₁₅	6	-2	0	0	-3	0	0	1	0	0	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	0	-ζ₉⁷-ζ₉²	-ζ₉⁵-ζ₉⁴	-ζ₉⁸-ζ₉	orthogonal faithful
ρ₁₆	6	-2	0	0	-3	0	0	1	0	0	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	0	-ζ₉⁵-ζ₉⁴	-ζ₉⁸-ζ₉	-ζ₉⁷-ζ₉²	orthogonal faithful
ρ₁₇	6	2	0	0	-3	0	0	-1	0	0	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	0	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	orthogonal faithful

Permutation representations of He₃.D₆
►On 27 points - transitive group 27T124

Generators in S₂₇

(1 22 19)(2 14 11)(3 24 21)(4 16 13)(5 26 23)(6 18 15)(7 10 25)(8 20 17)(9 12 27)

(1 4 7)(2 5 8)(3 6 9)(10 22 16)(11 23 17)(12 24 18)(13 25 19)(14 26 20)(15 27 21)

(1 22 13)(2 11 20)(4 16 25)(5 23 14)(7 10 19)(8 17 26)(12 24 18)(15 21 27)

(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)

(1 4)(2 3)(5 9)(6 8)(10 16)(11 15)(12 14)(17 27)(18 26)(19 25)(20 24)(21 23)

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

G:=Group( (1,22,19)(2,14,11)(3,24,21)(4,16,13)(5,26,23)(6,18,15)(7,10,25)(8,20,17)(9,12,27), (1,4,7)(2,5,8)(3,6,9)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(15,27,21), (1,22,13)(2,11,20)(4,16,25)(5,23,14)(7,10,19)(8,17,26)(12,24,18)(15,21,27), (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), (1,4)(2,3)(5,9)(6,8)(10,16)(11,15)(12,14)(17,27)(18,26)(19,25)(20,24)(21,23) );

G=PermutationGroup([[(1,22,19),(2,14,11),(3,24,21),(4,16,13),(5,26,23),(6,18,15),(7,10,25),(8,20,17),(9,12,27)], [(1,4,7),(2,5,8),(3,6,9),(10,22,16),(11,23,17),(12,24,18),(13,25,19),(14,26,20),(15,27,21)], [(1,22,13),(2,11,20),(4,16,25),(5,23,14),(7,10,19),(8,17,26),(12,24,18),(15,21,27)], [(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)], [(1,4),(2,3),(5,9),(6,8),(10,16),(11,15),(12,14),(17,27),(18,26),(19,25),(20,24),(21,23)]])

G:=TransitiveGroup(27,124);

►On 27 points - transitive group 27T132

Generators in S₂₇

(1 7 4)(2 8 5)(3 9 6)(10 22 16)(12 24 18)(14 26 20)

(1 4 7)(2 5 8)(3 6 9)(10 22 16)(11 23 17)(12 24 18)(13 25 19)(14 26 20)(15 27 21)

(1 16 25)(2 20 11)(3 24 15)(4 10 19)(5 14 23)(6 18 27)(7 22 13)(8 26 17)(9 12 21)

(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)

(1 3)(4 9)(5 8)(6 7)(10 15)(11 14)(12 13)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)

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

G:=Group( (1,7,4)(2,8,5)(3,9,6)(10,22,16)(12,24,18)(14,26,20), (1,4,7)(2,5,8)(3,6,9)(10,22,16)(11,23,17)(12,24,18)(13,25,19)(14,26,20)(15,27,21), (1,16,25)(2,20,11)(3,24,15)(4,10,19)(5,14,23)(6,18,27)(7,22,13)(8,26,17)(9,12,21), (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), (1,3)(4,9)(5,8)(6,7)(10,15)(11,14)(12,13)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22) );

G=PermutationGroup([[(1,7,4),(2,8,5),(3,9,6),(10,22,16),(12,24,18),(14,26,20)], [(1,4,7),(2,5,8),(3,6,9),(10,22,16),(11,23,17),(12,24,18),(13,25,19),(14,26,20),(15,27,21)], [(1,16,25),(2,20,11),(3,24,15),(4,10,19),(5,14,23),(6,18,27),(7,22,13),(8,26,17),(9,12,21)], [(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)], [(1,3),(4,9),(5,8),(6,7),(10,15),(11,14),(12,13),(16,27),(17,26),(18,25),(19,24),(20,23),(21,22)]])

G:=TransitiveGroup(27,132);

Matrix representation of He₃.D₆ ►in GL₆(𝔽₁₉)

0	0	1	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
0	1	0	0	0	0

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	0	0	18	1
0	0	0	0	18	0

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

18	4	4	16	18	4
15	3	3	1	15	3
18	4	16	18	16	18
15	3	1	15	1	15
4	16	4	16	16	18
3	1	3	1	1	15

16	4	16	4	18	16
1	3	1	3	15	1
16	4	4	18	4	18
1	3	3	15	3	15
18	16	4	18	18	16
15	1	3	15	15	1

G:=sub<GL(6,GF(19))| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[18,18,0,0,0,0,1,0,0,0,0,0,0,0,18,18,0,0,0,0,1,0,0,0,0,0,0,0,18,18,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,18,18,18,18,0,0,0,0,1,0,0,0,0,0],[18,15,18,15,4,3,4,3,4,3,16,1,4,3,16,1,4,3,16,1,18,15,16,1,18,15,16,1,16,1,4,3,18,15,18,15],[16,1,16,1,18,15,4,3,4,3,16,1,16,1,4,3,4,3,4,3,18,15,18,15,18,15,4,3,18,15,16,1,18,15,16,1] >;

He₃.D₆ in GAP, Magma, Sage, TeX

{\rm He}_3.D_6

% in TeX

G:=Group("He3.D6");

// GroupNames label

G:=SmallGroup(324,40);

// by ID

G=gap.SmallGroup(324,40);

# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,1052,986,579,303,5404,1090,382,3899]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=e^2=1,d^6=e*b*e=b^-1,a*b=b*a,c*a*c^-1=e*a*e=a*b^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=e*c*e=a^-1*c^-1,e*d*e=b*d^5>;

// generators/relations

Export

Subgroup lattice of He₃.D₆ in TeX
Character table of He₃.D₆ in TeX

G = He3.D6 order 324 = 22·34

1st non-split extension by He3 of D6 acting faithfully

G = He₃.D₆ order 324 = 2²·3⁴

1^st non-split extension by He₃ of D₆ acting faithfully