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## G = He3⋊3D4order 216 = 23·33

### 2nd semidirect product of He3 and D4 acting via D4/C2=C22

Aliases: He33D4, C32⋊D12, C6.20S32, (C3×C6).5D6, C3⋊Dic33S3, C32⋊C121C2, C322(C3⋊D4), C2.5(C32⋊D6), C3.3(C3⋊D12), (C2×He3).5C22, (C2×C3⋊S3)⋊2S3, (C2×C32⋊C6)⋊2C2, (C2×He3⋊C2)⋊1C2, SmallGroup(216,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — He3⋊3D4
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — He3⋊3D4
 Lower central He3 — C2×He3 — He3⋊3D4
 Upper central C1 — C2

Generators and relations for He33D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 368 in 66 conjugacy classes, 16 normal (all 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, C3⋊Dic3, S3×C6, C2×C3⋊S3, C32⋊C6, He3⋊C2, C2×He3, D6⋊S3, C3⋊D12, C32⋊C12, C2×C32⋊C6, C2×He3⋊C2, He33D4
Quotients: C1, C2, C22, S3, D4, D6, D12, C3⋊D4, S32, C3⋊D12, C32⋊D6, He33D4

Character table of He33D4

 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 0 2 2 -1 2 -1 0 2 -1 2 -1 0 -1 -1 0 0 0 orthogonal lifted from S3 ρ7 2 2 0 0 2 2 -1 -1 -2 2 2 -1 -1 0 0 0 0 1 1 orthogonal lifted from D6 ρ8 2 2 0 0 2 2 -1 -1 2 2 2 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ9 2 2 0 -2 2 -1 2 -1 0 2 -1 2 -1 0 1 1 0 0 0 orthogonal lifted from D6 ρ10 2 -2 0 0 2 2 -1 -1 0 -2 -2 1 1 0 0 0 0 √3 -√3 orthogonal lifted from D12 ρ11 2 -2 0 0 2 2 -1 -1 0 -2 -2 1 1 0 0 0 0 -√3 √3 orthogonal lifted from D12 ρ12 2 -2 0 0 2 -1 2 -1 0 -2 1 -2 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 0 √-3 -√-3 0 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 C3⋊D12 ρ15 4 4 0 0 4 -2 -2 1 0 4 -2 -2 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ16 6 6 2 0 -3 0 0 0 0 -3 0 0 0 -1 0 0 -1 0 0 orthogonal lifted from C32⋊D6 ρ17 6 6 -2 0 -3 0 0 0 0 -3 0 0 0 1 0 0 1 0 0 orthogonal lifted from C32⋊D6 ρ18 6 -6 0 0 -3 0 0 0 0 3 0 0 0 -√-3 0 0 √-3 0 0 complex faithful ρ19 6 -6 0 0 -3 0 0 0 0 3 0 0 0 √-3 0 0 -√-3 0 0 complex faithful

Smallest permutation representation of He33D4
On 36 points
Generators in S36
(1 16 7)(2 13 8)(3 14 5)(4 15 6)(9 29 18)(10 30 19)(11 31 20)(12 32 17)(21 27 33)(22 28 34)(23 25 35)(24 26 36)
(1 21 29)(2 30 22)(3 23 31)(4 32 24)(5 35 11)(6 12 36)(7 33 9)(8 10 34)(13 19 28)(14 25 20)(15 17 26)(16 27 18)
(5 11 35)(6 36 12)(7 9 33)(8 34 10)(13 19 28)(14 25 20)(15 17 26)(16 27 18)
(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 13)(6 16)(7 15)(8 14)(9 26)(10 25)(11 28)(12 27)(17 33)(18 36)(19 35)(20 34)(21 32)(22 31)(23 30)(24 29)

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

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

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

He33D4 is a maximal subgroup of   C12.84S32  C12.91S32  C12.S32  C3⋊S3⋊D12  C62.8D6  C62⋊D6  C622D6
He33D4 is a maximal quotient of   He33D8  He34SD16  He35SD16  He33Q16  C62.D6  C62.4D6  C62.5D6

Matrix representation of He33D4 in GL6(𝔽13)

 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0
,
 12 1 0 0 0 0 12 0 0 0 0 0 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 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 4 11 0 0 0 0 2 9 0 0 0 0 0 0 4 11 0 0 0 0 2 9 0 0 0 0 0 0 4 11 0 0 0 0 2 9
,
 2 9 0 0 0 0 4 11 0 0 0 0 0 0 0 0 2 9 0 0 0 0 4 11 0 0 2 9 0 0 0 0 4 11 0 0

G:=sub<GL(6,GF(13))| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[4,2,0,0,0,0,11,9,0,0,0,0,0,0,4,2,0,0,0,0,11,9,0,0,0,0,0,0,4,2,0,0,0,0,11,9],[2,4,0,0,0,0,9,11,0,0,0,0,0,0,0,0,2,4,0,0,0,0,9,11,0,0,2,4,0,0,0,0,9,11,0,0] >;

He33D4 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_3D_4
% in TeX

G:=Group("He3:3D4");
// GroupNames label

G:=SmallGroup(216,37);
// by ID

G=gap.SmallGroup(216,37);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,201,1444,382,5189,2603]);
// 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,a*d=d*a,e*a*e=a^-1,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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