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G = He35D6order 324 = 22·34

1st semidirect product of He3 and D6 acting via D6/C3=C22

non-abelian, supersoluble, monomial, rational

Aliases: He35D6, C334D6, C321S32, C32⋊C6⋊S3, C33⋊C22S3, C31(C32⋊D6), (C3×He3)⋊3C22, He34S32C2, He35S32C2, C3⋊S3⋊(C3⋊S3), C32⋊(C2×C3⋊S3), (C3×C3⋊S3)⋊2S3, C3.3(S3×C3⋊S3), (C3×C32⋊C6)⋊3C2, SmallGroup(324,121)

Series: Derived Chief Lower central Upper central

C1C3C3×He3 — He35D6
C1C3C32C33C3×He3C3×C32⋊C6 — He35D6
C3×He3 — He35D6
C1

Generators and relations for He35D6
 G = < a,b,c,d,e | a3=b3=c3=d6=e2=1, ab=ba, cac-1=ab-1, dad-1=eae=a-1, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1122 in 133 conjugacy classes, 24 normal (15 characteristic)
C1, C2 [×3], C3 [×2], C3 [×9], C22, S3 [×17], C6 [×6], C32 [×2], C32 [×3], C32 [×10], D6 [×6], C3×S3 [×17], C3⋊S3, C3⋊S3 [×9], C3×C6, He3 [×3], He3 [×3], C33 [×2], C33, S32 [×6], C2×C3⋊S3, C32⋊C6 [×3], C32⋊C6 [×3], He3⋊C2 [×6], S3×C32, C3×C3⋊S3, C3×C3⋊S3 [×4], C33⋊C2, C3×He3, C32⋊D6 [×3], S3×C3⋊S3, C324D6, C3×C32⋊C6, He34S3, He35S3, He35D6
Quotients: C1, C2 [×3], C22, S3 [×5], D6 [×5], C3⋊S3, S32 [×4], C2×C3⋊S3, C32⋊D6, S3×C3⋊S3, He35D6

Character table of He35D6

 class 12A2B2C3A3B3C3D3E3F3G3H3I3J3K6A6B6C6D6E6F
 size 192727224666612121212181818185454
ρ1111111111111111111111    trivial
ρ21-11-111111111111-1-1-1-1-11    linear of order 2
ρ31-1-1111111111111-1-1-1-11-1    linear of order 2
ρ411-1-1111111111111111-1-1    linear of order 2
ρ52-2002-1-1-1-122-1-12-11-21100    orthogonal lifted from D6
ρ62200222-1-1-12-12-1-1-1-1-1200    orthogonal lifted from S3
ρ7200-2222222-1-1-1-1-1000010    orthogonal lifted from D6
ρ822002-1-12-1-12-1-1-12-1-12-100    orthogonal lifted from S3
ρ92002222222-1-1-1-1-10000-10    orthogonal lifted from S3
ρ1022002-1-1-12-122-1-1-12-1-1-100    orthogonal lifted from S3
ρ1122002-1-1-1-122-1-12-1-12-1-100    orthogonal lifted from S3
ρ122-2002-1-12-1-12-1-1-1211-2100    orthogonal lifted from D6
ρ132-200222-1-1-12-12-1-1111-200    orthogonal lifted from D6
ρ142-2002-1-1-12-122-1-1-1-211100    orthogonal lifted from D6
ρ1540004-2-2-2-24-211-21000000    orthogonal lifted from S32
ρ1640004-2-24-2-2-2111-2000000    orthogonal lifted from S32
ρ174000444-2-2-2-21-211000000    orthogonal lifted from S32
ρ1840004-2-2-24-2-2-2111000000    orthogonal lifted from S32
ρ1960-20-36-300000000000001    orthogonal lifted from C32⋊D6
ρ206020-36-30000000000000-1    orthogonal lifted from C32⋊D6
ρ2112000-6-6300000000000000    orthogonal faithful

Permutation representations of He35D6
On 18 points - transitive group 18T133
Generators in S18
(1 11 15)(2 16 12)(3 7 17)(4 18 8)(5 9 13)(6 14 10)
(1 5 3)(2 4 6)(7 11 9)(8 10 12)(13 17 15)(14 16 18)
(7 11 9)(8 12 10)(13 15 17)(14 16 18)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 6)(2 5)(3 4)(7 8)(9 12)(10 11)(13 16)(14 15)(17 18)

G:=sub<Sym(18)| (1,11,15)(2,16,12)(3,7,17)(4,18,8)(5,9,13)(6,14,10), (1,5,3)(2,4,6)(7,11,9)(8,10,12)(13,17,15)(14,16,18), (7,11,9)(8,12,10)(13,15,17)(14,16,18), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,8)(9,12)(10,11)(13,16)(14,15)(17,18)>;

G:=Group( (1,11,15)(2,16,12)(3,7,17)(4,18,8)(5,9,13)(6,14,10), (1,5,3)(2,4,6)(7,11,9)(8,10,12)(13,17,15)(14,16,18), (7,11,9)(8,12,10)(13,15,17)(14,16,18), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,8)(9,12)(10,11)(13,16)(14,15)(17,18) );

G=PermutationGroup([(1,11,15),(2,16,12),(3,7,17),(4,18,8),(5,9,13),(6,14,10)], [(1,5,3),(2,4,6),(7,11,9),(8,10,12),(13,17,15),(14,16,18)], [(7,11,9),(8,12,10),(13,15,17),(14,16,18)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,16),(14,15),(17,18)])

G:=TransitiveGroup(18,133);

On 18 points - transitive group 18T139
Generators in S18
(7 9 11)(8 12 10)(13 17 15)(14 16 18)
(1 5 3)(2 4 6)(7 11 9)(8 10 12)(13 17 15)(14 16 18)
(1 11 15)(2 12 16)(3 7 17)(4 8 18)(5 9 13)(6 10 14)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 6)(2 5)(3 4)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)

G:=sub<Sym(18)| (7,9,11)(8,12,10)(13,17,15)(14,16,18), (1,5,3)(2,4,6)(7,11,9)(8,10,12)(13,17,15)(14,16,18), (1,11,15)(2,12,16)(3,7,17)(4,8,18)(5,9,13)(6,10,14), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)>;

G:=Group( (7,9,11)(8,12,10)(13,17,15)(14,16,18), (1,5,3)(2,4,6)(7,11,9)(8,10,12)(13,17,15)(14,16,18), (1,11,15)(2,12,16)(3,7,17)(4,8,18)(5,9,13)(6,10,14), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13) );

G=PermutationGroup([(7,9,11),(8,12,10),(13,17,15),(14,16,18)], [(1,5,3),(2,4,6),(7,11,9),(8,10,12),(13,17,15),(14,16,18)], [(1,11,15),(2,12,16),(3,7,17),(4,8,18),(5,9,13),(6,10,14)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,6),(2,5),(3,4),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13)])

G:=TransitiveGroup(18,139);

On 27 points - transitive group 27T117
Generators in S27
(1 10 13)(2 14 11)(3 12 15)(4 24 17)(5 18 25)(6 26 19)(7 20 27)(8 22 21)(9 16 23)
(1 5 8)(2 9 6)(3 7 4)(10 18 22)(11 23 19)(12 20 24)(13 25 21)(14 16 26)(15 27 17)
(1 3 2)(4 6 8)(5 7 9)(10 20 26)(11 21 27)(12 16 22)(13 17 23)(14 18 24)(15 19 25)
(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 2)(5 9)(6 8)(10 11)(12 15)(13 14)(16 25)(17 24)(18 23)(19 22)(20 27)(21 26)

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

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

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

G:=TransitiveGroup(27,117);

On 27 points - transitive group 27T127
Generators in S27
(1 16 19)(2 20 17)(3 18 21)(4 11 14)(5 15 12)(6 13 10)
(1 19 16)(2 17 20)(3 21 18)(4 11 14)(5 15 12)(6 13 10)(7 26 23)(8 24 27)(9 22 25)
(1 5 9)(2 6 7)(3 4 8)(10 23 20)(11 24 21)(12 25 16)(13 26 17)(14 27 18)(15 22 19)
(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 5)(2 4)(3 6)(7 8)(10 18)(11 17)(12 16)(13 21)(14 20)(15 19)(23 27)(24 26)

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

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

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

G:=TransitiveGroup(27,127);

Matrix representation of He35D6 in GL10(ℤ)

-10-10000000
0-10-1000000
1000000000
0100000000
0000000010
0000100000
0000000100
0000000001
0000010000
0000001000
,
1000000000
0100000000
0010000000
0001000000
000000-1000
000000000-1
000010-1000
0000000-110
0000000-100
000001000-1
,
1000000000
0100000000
0010000000
0001000000
0000100000
000000000-1
0000001000
00000000-10
00000001-10
000001000-1
,
-1100000000
-1000000000
1-11-1000000
1010000000
000000-1000
0000000-100
0000-100000
00000-10000
000000000-1
00000000-10
,
-1000000000
-1100000000
1010000000
1-11-1000000
0000-100000
00000000-10
000000-1000
000000000-1
00000-10000
0000000-100

G:=sub<GL(10,Integers())| [-1,0,1,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,-1,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,-1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,-1,0,0,0,-1],[-1,-1,1,1,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0],[-1,-1,1,1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0] >;

He35D6 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_5D_6
% in TeX

G:=Group("He3:5D6");
// GroupNames label

G:=SmallGroup(324,121);
// by ID

G=gap.SmallGroup(324,121);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,80,297,2164,1096,7781,3899]);
// Polycyclic

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

Export

Character table of He35D6 in TeX

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