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## G = He3⋊6D4order 216 = 23·33

### 1st semidirect product of He3 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — He3⋊6D4
 Chief series C1 — C3 — C32 — C3×C6 — C2×He3 — C2×C32⋊C6 — He3⋊6D4
 Lower central C32 — C3×C6 — He3⋊6D4
 Upper central C1 — C2 — C22

Generators and relations for He36D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, cac-1=ab-1, dad-1=eae=a-1, bc=cb, dbd-1=ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 270 in 66 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, D4, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊D4, C3×D4, He3, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C62, C32⋊C6, C2×He3, C2×He3, C3×C3⋊D4, C327D4, C32⋊C12, C2×C32⋊C6, C22×He3, He36D4
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, C3×S3, C3⋊D4, C3×D4, S3×C6, C32⋊C6, C3×C3⋊D4, C2×C32⋊C6, He36D4

Smallest permutation representation of He36D4
On 36 points
Generators in S36
(1 16 11)(2 12 13)(3 14 9)(4 10 15)(5 31 26)(6 27 32)(7 29 28)(8 25 30)(17 36 22)(18 23 33)(19 34 24)(20 21 35)
(1 8 21)(2 22 5)(3 6 23)(4 24 7)(9 32 18)(10 19 29)(11 30 20)(12 17 31)(13 36 26)(14 27 33)(15 34 28)(16 25 35)
(9 18 32)(10 19 29)(11 20 30)(12 17 31)(13 26 36)(14 27 33)(15 28 34)(16 25 35)
(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 28)(29 30 31 32)(33 34 35 36)
(2 4)(5 24)(6 23)(7 22)(8 21)(9 14)(10 13)(11 16)(12 15)(17 28)(18 27)(19 26)(20 25)(29 36)(30 35)(31 34)(32 33)

G:=sub<Sym(36)| (1,16,11)(2,12,13)(3,14,9)(4,10,15)(5,31,26)(6,27,32)(7,29,28)(8,25,30)(17,36,22)(18,23,33)(19,34,24)(20,21,35), (1,8,21)(2,22,5)(3,6,23)(4,24,7)(9,32,18)(10,19,29)(11,30,20)(12,17,31)(13,36,26)(14,27,33)(15,34,28)(16,25,35), (9,18,32)(10,19,29)(11,20,30)(12,17,31)(13,26,36)(14,27,33)(15,28,34)(16,25,35), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (2,4)(5,24)(6,23)(7,22)(8,21)(9,14)(10,13)(11,16)(12,15)(17,28)(18,27)(19,26)(20,25)(29,36)(30,35)(31,34)(32,33)>;

G:=Group( (1,16,11)(2,12,13)(3,14,9)(4,10,15)(5,31,26)(6,27,32)(7,29,28)(8,25,30)(17,36,22)(18,23,33)(19,34,24)(20,21,35), (1,8,21)(2,22,5)(3,6,23)(4,24,7)(9,32,18)(10,19,29)(11,30,20)(12,17,31)(13,36,26)(14,27,33)(15,34,28)(16,25,35), (9,18,32)(10,19,29)(11,20,30)(12,17,31)(13,26,36)(14,27,33)(15,28,34)(16,25,35), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (2,4)(5,24)(6,23)(7,22)(8,21)(9,14)(10,13)(11,16)(12,15)(17,28)(18,27)(19,26)(20,25)(29,36)(30,35)(31,34)(32,33) );

G=PermutationGroup([[(1,16,11),(2,12,13),(3,14,9),(4,10,15),(5,31,26),(6,27,32),(7,29,28),(8,25,30),(17,36,22),(18,23,33),(19,34,24),(20,21,35)], [(1,8,21),(2,22,5),(3,6,23),(4,24,7),(9,32,18),(10,19,29),(11,30,20),(12,17,31),(13,36,26),(14,27,33),(15,34,28),(16,25,35)], [(9,18,32),(10,19,29),(11,20,30),(12,17,31),(13,26,36),(14,27,33),(15,28,34),(16,25,35)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36)], [(2,4),(5,24),(6,23),(7,22),(8,21),(9,14),(10,13),(11,16),(12,15),(17,28),(18,27),(19,26),(20,25),(29,36),(30,35),(31,34),(32,33)]])

He36D4 is a maximal subgroup of   C62.8D6  C62.9D6  C62⋊D6  C622D6  C62.36D6  D4×C32⋊C6  C62.13D6
He36D4 is a maximal quotient of   C62.19D6  C62.21D6  He38SD16  He36D8  He36Q16  He310SD16  C623C12

31 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4 6A 6B 6C 6D 6E 6F ··· 6P 6Q 6R 12A 12B order 1 2 2 2 3 3 3 3 3 3 4 6 6 6 6 6 6 ··· 6 6 6 12 12 size 1 1 2 18 2 3 3 6 6 6 18 2 2 2 3 3 6 ··· 6 18 18 18 18

31 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 6 6 6 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×C3⋊D4 C32⋊C6 C2×C32⋊C6 He3⋊6D4 kernel He3⋊6D4 C32⋊C12 C2×C32⋊C6 C22×He3 C32⋊7D4 C3⋊Dic3 C2×C3⋊S3 C62 C62 He3 C3×C6 C2×C6 C32 C32 C6 C3 C22 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 2 2 2 4 1 1 2

Matrix representation of He36D4 in GL8(𝔽13)

 1 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 1 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
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0
,
 3 0 0 0 0 0 0 0 0 3 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 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0
,
 6 4 0 0 0 0 0 0 7 7 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0
,
 1 0 0 0 0 0 0 0 10 12 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[6,7,0,0,0,0,0,0,4,7,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0],[1,10,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;

He36D4 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_6D_4
% in TeX

G:=Group("He3:6D4");
// GroupNames label

G:=SmallGroup(216,60);
// by ID

G=gap.SmallGroup(216,60);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,1444,736,5189]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=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=e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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