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## G = He3⋊2D4order 216 = 23·33

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

Aliases: He32D4, C6.18S32, (C3×C6).3D6, He33C41C2, C321(C3⋊D4), C2.4(C32⋊D6), C3.1(D6⋊S3), (C2×He3).3C22, (C2×C3⋊S3)⋊1S3, (C2×C32⋊C6)⋊1C2, SmallGroup(216,35)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — He3⋊2D4
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — He3⋊2D4
 Lower central He3 — C2×He3 — He3⋊2D4
 Upper central C1 — C2

Generators and relations for He32D4
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, bd=db, ebe=b-1, dcd-1=c-1, ce=ec, ede=d-1 >

Subgroups: 352 in 62 conjugacy classes, 16 normal (8 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, D4, C32, C32, Dic3, C12, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, D12, C3⋊D4, He3, C3×Dic3, S3×C6, C2×C3⋊S3, C32⋊C6, C2×He3, C3⋊D12, He33C4, C2×C32⋊C6, He32D4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊D4, S32, D6⋊S3, C32⋊D6, He32D4

Character table of He32D4

 class 1 2A 2B 2C 3A 3B 3C 3D 4 6A 6B 6C 6D 6E 6F 6G 6H 12A 12B size 1 1 18 18 2 6 6 12 18 2 6 6 12 18 18 18 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 1 1 -1 1 1 1 1 -1 1 1 1 1 -1 1 1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ5 2 -2 0 0 2 2 2 2 0 -2 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 2 2 2 0 2 2 -1 -1 0 2 2 -1 -1 0 -1 -1 0 0 0 orthogonal lifted from S3 ρ7 2 2 0 2 2 -1 2 -1 0 2 -1 2 -1 -1 0 0 -1 0 0 orthogonal lifted from S3 ρ8 2 2 -2 0 2 2 -1 -1 0 2 2 -1 -1 0 1 1 0 0 0 orthogonal lifted from D6 ρ9 2 2 0 -2 2 -1 2 -1 0 2 -1 2 -1 1 0 0 1 0 0 orthogonal lifted from D6 ρ10 2 -2 0 0 2 2 -1 -1 0 -2 -2 1 1 0 √-3 -√-3 0 0 0 complex lifted from C3⋊D4 ρ11 2 -2 0 0 2 -1 2 -1 0 -2 1 -2 1 -√-3 0 0 √-3 0 0 complex lifted from C3⋊D4 ρ12 2 -2 0 0 2 2 -1 -1 0 -2 -2 1 1 0 -√-3 √-3 0 0 0 complex lifted from C3⋊D4 ρ13 2 -2 0 0 2 -1 2 -1 0 -2 1 -2 1 √-3 0 0 -√-3 0 0 complex lifted from C3⋊D4 ρ14 4 4 0 0 4 -2 -2 1 0 4 -2 -2 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ15 4 -4 0 0 4 -2 -2 1 0 -4 2 2 -1 0 0 0 0 0 0 symplectic lifted from D6⋊S3, Schur index 2 ρ16 6 6 0 0 -3 0 0 0 -2 -3 0 0 0 0 0 0 0 1 1 orthogonal lifted from C32⋊D6 ρ17 6 6 0 0 -3 0 0 0 2 -3 0 0 0 0 0 0 0 -1 -1 orthogonal lifted from C32⋊D6 ρ18 6 -6 0 0 -3 0 0 0 0 3 0 0 0 0 0 0 0 √3 -√3 orthogonal faithful ρ19 6 -6 0 0 -3 0 0 0 0 3 0 0 0 0 0 0 0 -√3 √3 orthogonal faithful

Smallest permutation representation of He32D4
On 36 points
Generators in S36
(1 30 13)(2 14 31)(3 32 15)(4 16 29)(5 18 9)(6 10 19)(7 20 11)(8 12 17)(21 34 28)(22 25 35)(23 36 26)(24 27 33)
(1 5 24)(2 6 21)(3 7 22)(4 8 23)(9 33 13)(10 34 14)(11 35 15)(12 36 16)(17 26 29)(18 27 30)(19 28 31)(20 25 32)
(9 13 33)(10 34 14)(11 15 35)(12 36 16)(17 29 26)(18 27 30)(19 31 28)(20 25 32)
(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)
(1 4)(2 3)(5 23)(6 22)(7 21)(8 24)(9 36)(10 35)(11 34)(12 33)(13 16)(14 15)(17 27)(18 26)(19 25)(20 28)(29 30)(31 32)

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

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

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

He32D4 is a maximal subgroup of   He32SD16  He3⋊D8  C12⋊S3⋊S3  C12.91S32  C12.86S32  C62.9D6  C62⋊D6
He32D4 is a maximal quotient of   He33SD16  He32D8  He32Q16  C62.3D6  C62.4D6

Matrix representation of He32D4 in GL6(𝔽13)

 0 0 0 0 12 1 3 3 0 0 11 12 0 0 0 0 4 0 1 0 0 0 4 0 0 0 1 0 10 0 0 0 0 1 10 0
,
 0 12 0 0 0 0 1 12 0 0 0 0 9 0 12 12 0 0 0 4 1 0 0 0 3 0 0 0 12 12 0 10 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 9 9 12 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 3 3 0 0 12 12
,
 3 7 0 0 0 0 6 10 0 0 0 0 10 8 0 0 10 7 5 3 0 0 6 3 10 5 10 7 0 0 8 3 6 3 0 0
,
 7 3 0 0 0 0 10 6 0 0 0 0 3 5 0 0 3 6 12 1 0 0 3 10 3 8 3 6 0 0 4 9 3 10 0 0

G:=sub<GL(6,GF(13))| [0,3,0,1,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,11,4,4,10,10,1,12,0,0,0,0],[0,1,9,0,3,0,12,12,0,4,0,10,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,0,9,0,0,3,0,1,9,0,0,3,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[3,6,10,5,10,8,7,10,8,3,5,3,0,0,0,0,10,6,0,0,0,0,7,3,0,0,10,6,0,0,0,0,7,3,0,0],[7,10,3,12,3,4,3,6,5,1,8,9,0,0,0,0,3,3,0,0,0,0,6,10,0,0,3,3,0,0,0,0,6,10,0,0] >;

He32D4 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_2D_4
% in TeX

G:=Group("He3:2D4");
// GroupNames label

G:=SmallGroup(216,35);
// by ID

G=gap.SmallGroup(216,35);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,201,1444,382,5189,2603]);
// 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,b*d=d*b,e*b*e=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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