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G = He34S3order 162 = 2·34

1st semidirect product of He3 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: He34S3, C334C6, C334S3, C3⋊(C32⋊C6), (C3×He3)⋊3C2, C323(C3×S3), C321(C3⋊S3), C33⋊C22C3, C3.2(C3×C3⋊S3), SmallGroup(162,40)

Series: Derived Chief Lower central Upper central

C1C33 — He34S3
C1C3C32C33C3×He3 — He34S3
C33 — He34S3
C1

Generators and relations for He34S3
 G = < a,b,c,d,e | a3=b3=c3=d3=e2=1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 364 in 67 conjugacy classes, 18 normal (9 characteristic)
C1, C2, C3, C3 [×3], C3 [×8], S3 [×7], C6, C32 [×2], C32 [×3], C32 [×13], C3×S3 [×4], C3⋊S3 [×7], He3 [×3], He3 [×3], C33 [×2], C33, C32⋊C6 [×3], C3×C3⋊S3, C33⋊C2, C3×He3, He34S3
Quotients: C1, C2, C3, S3 [×4], C6, C3×S3 [×4], C3⋊S3, C32⋊C6 [×3], C3×C3⋊S3, He34S3

Character table of He34S3

 class 123A3B3C3D3E3F3G3H3I3J3K3L3M3N3O3P3Q6A6B
 size 127222233666666666662727
ρ1111111111111111111111    trivial
ρ21-111111111111111111-1-1    linear of order 2
ρ3111111ζ32ζ3ζ32ζ321ζ32ζ321ζ3ζ3ζ3ζ31ζ32ζ3    linear of order 3
ρ41-11111ζ3ζ32ζ3ζ31ζ3ζ31ζ32ζ32ζ32ζ321ζ65ζ6    linear of order 6
ρ51-11111ζ32ζ3ζ32ζ321ζ32ζ321ζ3ζ3ζ3ζ31ζ6ζ65    linear of order 6
ρ6111111ζ3ζ32ζ3ζ31ζ3ζ31ζ32ζ32ζ32ζ321ζ3ζ32    linear of order 3
ρ720-1-12-122-1-1-1-122-1-1-12-100    orthogonal lifted from S3
ρ820222222-12-1-1-1-12-1-1-1-100    orthogonal lifted from S3
ρ920-1-12-122-1-122-1-1-1-12-1-100    orthogonal lifted from S3
ρ1020-1-12-1222-1-1-1-1-1-12-1-1200    orthogonal lifted from S3
ρ1120-1-12-1-1--3-1+-3ζ6ζ6-1ζ6-1--32ζ65ζ65ζ65-1+-3-100    complex lifted from C3×S3
ρ1220-1-12-1-1+-3-1--3ζ65ζ65-1ζ65-1+-32ζ6ζ6ζ6-1--3-100    complex lifted from C3×S3
ρ13202222-1--3-1+-3ζ6-1--3-1ζ6ζ6-1-1+-3ζ65ζ65ζ65-100    complex lifted from C3×S3
ρ1420-1-12-1-1+-3-1--3-1+-3ζ65-1ζ65ζ65-1ζ6-1--3ζ6ζ6200    complex lifted from C3×S3
ρ1520-1-12-1-1--3-1+-3ζ6ζ62-1--3ζ6-1ζ65ζ65-1+-3ζ65-100    complex lifted from C3×S3
ρ1620-1-12-1-1--3-1+-3-1--3ζ6-1ζ6ζ6-1ζ65-1+-3ζ65ζ65200    complex lifted from C3×S3
ρ17202222-1+-3-1--3ζ65-1+-3-1ζ65ζ65-1-1--3ζ6ζ6ζ6-100    complex lifted from C3×S3
ρ1820-1-12-1-1+-3-1--3ζ65ζ652-1+-3ζ65-1ζ6ζ6-1--3ζ6-100    complex lifted from C3×S3
ρ19606-3-3-3000000000000000    orthogonal lifted from C32⋊C6
ρ2060-36-3-3000000000000000    orthogonal lifted from C32⋊C6
ρ2160-3-3-36000000000000000    orthogonal lifted from C32⋊C6

Permutation representations of He34S3
On 27 points - transitive group 27T61
Generators in S27
(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 26 8)(2 27 9)(3 25 7)(4 19 23)(5 20 24)(6 21 22)(10 18 15)(11 16 13)(12 17 14)
(2 27 9)(3 7 25)(4 19 23)(5 24 20)(10 18 15)(11 13 16)
(1 14 6)(2 15 4)(3 13 5)(7 16 24)(8 17 22)(9 18 23)(10 19 27)(11 20 25)(12 21 26)
(2 3)(4 13)(5 15)(6 14)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(16 19)(17 21)(18 20)

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

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

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

G:=TransitiveGroup(27,61);

On 27 points - transitive group 27T71
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 3 2)(4 6 5)(7 9 8)(10 11 12)(13 14 15)(16 17 18)(19 21 20)(22 24 23)(25 27 26)
(1 19 10)(2 20 12)(3 21 11)(4 22 14)(5 23 13)(6 24 15)(7 25 17)(8 26 16)(9 27 18)
(1 4 7)(2 5 8)(3 6 9)(10 14 17)(11 15 18)(12 13 16)(19 22 25)(20 23 26)(21 24 27)
(2 3)(4 7)(5 9)(6 8)(11 12)(13 18)(14 17)(15 16)(20 21)(22 25)(23 27)(24 26)

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

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

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

G:=TransitiveGroup(27,71);

He34S3 is a maximal subgroup of
S3×C32⋊C6  He35D6  He36D6  He3⋊D9  (C3×He3)⋊S3  (C3×He3).S3  C32⋊C96S3  He32D9  C34⋊S3  (C3×He3)⋊C6  He33D9  C344C6  C9⋊He32C2  (C32×C9)⋊C6  C345C6  C324D9⋊C3  He3⋊C33S3  C347S3  He3.(C3⋊S3)  C3⋊(He3⋊S3)  3+ 1+4⋊C2  C3410C6  C9○He33S3
He34S3 is a maximal quotient of
C334C12  C33⋊C18  C33⋊D9  He33D9  C343S3  C344C6  C9⋊He32C2  (C32×C9)⋊8S3  (C32×C9)⋊C6  C345S3  C345C6  He3.C3⋊S3  C324D9⋊C3  He3⋊C32S3  He3⋊C33S3  C3≀C3.S3  C3410C6

Matrix representation of He34S3 in GL8(𝔽7)

66000000
10000000
00006100
00115600
00116010
00116001
00662000
00062000
,
10000000
01000000
00610000
00600000
00600100
00016600
00200001
00050066
,
40000000
04000000
00100000
00010000
00116600
00001000
00000001
00550066
,
66000000
10000000
00100000
00010000
00001000
00000100
00000010
00000001
,
01000000
10000000
00010000
00100000
00110010
00660066
00661000
00006600

G:=sub<GL(8,GF(7))| [6,1,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1,1,1,6,0,0,0,0,1,1,1,6,6,0,0,6,5,6,6,2,2,0,0,1,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,6,6,0,2,0,0,0,1,0,0,1,0,5,0,0,0,0,0,6,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,1,6],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,1,0,0,5,0,0,0,1,1,0,0,5,0,0,0,0,6,1,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,1,6],[6,1,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,6,6,0,0,0,1,0,1,6,6,0,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,6,0,0,0,0,1,6,0,0,0,0,0,0,0,6,0,0] >;

He34S3 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_4S_3
% in TeX

G:=Group("He3:4S3");
// GroupNames label

G:=SmallGroup(162,40);
// by ID

G=gap.SmallGroup(162,40);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,182,457,723,2704]);
// Polycyclic

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

Export

Character table of He34S3 in TeX

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