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## G = He3.(C3×C6)  order 486 = 2·35

### 6th non-split extension by He3 of C3×C6 acting via C3×C6/C3=C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3.(C3×C6)
 Chief series C1 — C3 — C32 — He3 — C3×He3 — He3⋊C32 — He3.(C3×C6)
 Lower central He3 — He3.(C3×C6)
 Upper central C1 — C3 — C32

Generators and relations for He3.(C3×C6)
G = < a,b,c,d,e | a3=b3=c3=e6=1, d3=b, ab=ba, cac-1=ab-1, ad=da, eae-1=a-1b, bc=cb, bd=db, be=eb, dcd-1=a-1bc, ece-1=c-1, ede-1=b-1d >

Subgroups: 468 in 82 conjugacy classes, 21 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C18, C3×S3, C3×C6, C3×C9, He3, He3, 3- 1+2, C33, C33, S3×C9, He3⋊C2, C2×3- 1+2, S3×C32, He3⋊C3, He3⋊C3, C3×He3, C3×He3, C3×3- 1+2, He3.2C6, S3×3- 1+2, C3×He3⋊C2, He3⋊C32, He3.(C3×C6)
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C32⋊C6, S3×C32, C3×C32⋊C6, He3.(C3×C6)

Permutation representations of He3.(C3×C6)
On 27 points - transitive group 27T183
Generators in S27
(1 7 4)(2 8 5)(3 9 6)(19 22 25)(20 23 26)(21 24 27)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)
(1 20 12)(2 27 10)(3 25 17)(4 23 15)(5 21 13)(6 19 11)(7 26 18)(8 24 16)(9 22 14)
(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)
(2 5 8)(3 9 6)(10 21 16 27 13 24)(11 25 14 19 17 22)(12 20)(15 23)(18 26)

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

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

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

G:=TransitiveGroup(27,183);

On 27 points - transitive group 27T203
Generators in S27
(1 20 14)(2 21 15)(3 22 16)(4 23 17)(5 24 18)(6 25 10)(7 26 11)(8 27 12)(9 19 13)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)
(2 15 27)(3 22 10)(5 18 21)(6 25 13)(8 12 24)(9 19 16)(11 17 14)(20 23 26)
(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)
(2 5 8)(3 9 6)(10 19 13 22 16 25)(11 23)(12 27 18 24 15 21)(14 26)(17 20)

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

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

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

G:=TransitiveGroup(27,203);

34 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H ··· 3P 6A 6B 6C 6D 9A ··· 9F 18A ··· 18F order 1 2 3 3 3 3 3 3 3 3 ··· 3 6 6 6 6 9 ··· 9 18 ··· 18 size 1 9 1 1 3 3 6 6 6 18 ··· 18 9 9 27 27 9 ··· 9 27 ··· 27

34 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 6 6 9 type + + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3×S3 C32⋊C6 C3×C32⋊C6 He3.(C3×C6) kernel He3.(C3×C6) He3⋊C32 He3.2C6 C3×He3⋊C2 He3⋊C3 C3×He3 C3×3- 1+2 C3×C9 C33 C32 C3 C1 # reps 1 1 6 2 6 2 1 6 2 1 2 4

Matrix representation of He3.(C3×C6) in GL9(𝔽19)

 0 1 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 0 1 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 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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 0 0 0 0 0 0 0 0 7
,
 1 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 11 0
,
 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 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
,
 1 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 11 0 0 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 7 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 1 0

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

He3.(C3×C6) in GAP, Magma, Sage, TeX

{\rm He}_3.(C_3\times C_6)
% in TeX

G:=Group("He3.(C3xC6)");
// GroupNames label

G:=SmallGroup(486,130);
// by ID

G=gap.SmallGroup(486,130);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1520,867,873,8104,382]);
// Polycyclic

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

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