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## G = He3⋊C32order 243 = 35

### 3rd semidirect product of He3 and C32 acting via C32/C3=C3

p-group, metabelian, nilpotent (class 3), monomial

Aliases: He33C32, C32.15He3, C32.8C33, C33.14C32, (C3×He3)⋊6C3, (C3×C9)⋊3C32, C3.14(C3×He3), He3⋊C33C3, (C3×3- 1+2)⋊9C3, 3-Sylow(PSigmaL(3,64)), SmallGroup(243,58)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C32 — He3⋊C32
 Chief series C1 — C3 — C32 — C33 — C3×3- 1+2 — He3⋊C32
 Lower central C1 — C3 — C32 — He3⋊C32
 Upper central C1 — C3 — C33 — He3⋊C32
 Jennings C1 — C3 — C32 — He3⋊C32

Generators and relations for He3⋊C32
G = < a,b,c,d,e | a3=b3=c3=d3=e3=1, ab=ba, cac-1=dad-1=ab-1, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ac, ece-1=b-1c, de=ed >

Subgroups: 288 in 80 conjugacy classes, 33 normal (6 characteristic)
C1, C3, C3, C9, C32, C32, C32, C3×C9, He3, He3, 3- 1+2, C33, C33, He3⋊C3, C3×He3, C3×3- 1+2, He3⋊C32
Quotients: C1, C3, C32, He3, C33, C3×He3, He3⋊C32

Permutation representations of He3⋊C32
On 27 points - transitive group 27T96
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 26 14)(2 27 13)(3 25 15)(4 21 16)(5 19 18)(6 20 17)(7 23 12)(8 24 11)(9 22 10)
(1 23 17)(2 24 16)(3 22 18)(4 25 10)(5 26 12)(6 27 11)(7 20 13)(8 21 15)(9 19 14)
(4 6 5)(7 8 9)(10 11 12)(13 15 14)(19 20 21)(25 27 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), (1,26,14)(2,27,13)(3,25,15)(4,21,16)(5,19,18)(6,20,17)(7,23,12)(8,24,11)(9,22,10), (1,23,17)(2,24,16)(3,22,18)(4,25,10)(5,26,12)(6,27,11)(7,20,13)(8,21,15)(9,19,14), (4,6,5)(7,8,9)(10,11,12)(13,15,14)(19,20,21)(25,27,26)>;

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

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

G:=TransitiveGroup(27,96);

On 27 points - transitive group 27T97
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 14 10)(2 15 11)(3 13 12)(4 27 8)(5 25 9)(6 26 7)(16 23 19)(17 24 20)(18 22 21)
(1 15 3)(2 12 10)(4 7 5)(6 25 27)(8 26 9)(11 13 14)(16 19 23)(18 22 21)
(1 6 19)(2 27 24)(3 9 18)(4 17 11)(5 22 13)(7 23 10)(8 20 15)(12 25 21)(14 26 16)
(1 13 11)(2 14 12)(3 15 10)(4 6 5)(7 9 8)(16 21 24)(17 19 22)(18 20 23)(25 27 26)

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

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

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

G:=TransitiveGroup(27,97);

He3⋊C32 is a maximal subgroup of   He3.(C3×C6)  He3.(C3×S3)  He3⋊(C3×S3)

35 conjugacy classes

 class 1 3A 3B 3C ··· 3J 3K ··· 3AB 9A ··· 9F order 1 3 3 3 ··· 3 3 ··· 3 9 ··· 9 size 1 1 1 3 ··· 3 9 ··· 9 9 ··· 9

35 irreducible representations

 dim 1 1 1 1 3 9 type + image C1 C3 C3 C3 He3 He3⋊C32 kernel He3⋊C32 He3⋊C3 C3×He3 C3×3- 1+2 C32 C1 # reps 1 18 6 2 6 2

Matrix representation of He3⋊C32 in GL9(𝔽19)

 0 1 0 0 0 0 0 0 0 12 8 6 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 8 6 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 8 6 0 0 0 0 0 0 0 0 11
,
 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 7
,
 0 7 0 0 0 0 0 0 0 12 8 6 0 0 0 0 0 0 8 7 11 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 12 1 7 0 0 0 0 0 0 0 0 0 18 12 9 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 18 11 1
,
 0 0 0 11 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 18 11 1 0 0 0 0 0 0 0 0 0 8 18 4 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 8 7 11 0 11 0 0 0 0 0 0 0 8 18 4 0 0 0 0 0 0 18 11 1 0 0 0 0 0 0
,
 18 12 9 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 8 6 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 8 18 4 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 11

G:=sub<GL(9,GF(19))| [0,12,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,6,11,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,6,11,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,6,11],[7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7],[0,12,8,0,0,0,0,0,0,7,8,7,0,0,0,0,0,0,0,6,11,0,0,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,18,7,18,0,0,0,0,0,0,12,0,11,0,0,0,0,0,0,9,0,1],[0,0,0,0,0,0,0,8,18,0,0,0,0,0,0,11,18,11,0,0,0,0,0,0,0,4,1,11,0,18,0,0,0,0,0,0,0,7,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,8,1,8,0,0,0,0,0,0,18,0,7,0,0,0,0,0,0,4,0,11,0,0,0],[18,11,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,9,0,1,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,6,0,7,0,0,0,0,0,0,0,0,0,8,7,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,4,0,11] >;

He3⋊C32 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes C_3^2
% in TeX

G:=Group("He3:C3^2");
// GroupNames label

G:=SmallGroup(243,58);
// by ID

G=gap.SmallGroup(243,58);
# by ID

G:=PCGroup([5,-3,3,3,-3,-3,301,546,457,2163]);
// Polycyclic

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

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