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## G = He3⋊4D4order 216 = 23·33

### 1st semidirect product of He3 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — He3⋊4D4
 Chief series C1 — C3 — C32 — C3×C6 — C2×He3 — C2×C32⋊C6 — He3⋊4D4
 Lower central C32 — C3×C6 — He3⋊4D4
 Upper central C1 — C2 — C4

Generators and relations for He34D4
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, bd=db, ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 302 in 62 conjugacy classes, 21 normal (17 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4, C22 [×2], S3 [×4], C6, C6 [×5], D4, C32 [×2], C32, C12, C12 [×3], D6 [×4], C2×C6 [×2], C3×S3 [×2], C3⋊S3 [×2], C3×C6 [×2], C3×C6, D12 [×2], C3×D4, He3, C3×C12 [×2], C3×C12, S3×C6 [×2], C2×C3⋊S3 [×2], C32⋊C6 [×2], C2×He3, C3×D12, C12⋊S3, C4×He3, C2×C32⋊C6 [×2], He34D4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, C3×S3, D12, C3×D4, S3×C6, C32⋊C6, C3×D12, C2×C32⋊C6, He34D4

Smallest permutation representation of He34D4
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 22 30)(3 23 31)(4 24 32)(5 35 11)(6 36 12)(7 33 9)(8 34 10)(13 28 19)(14 25 20)(15 26 17)(16 27 18)
(5 11 35)(6 12 36)(7 9 33)(8 10 34)(13 28 19)(14 25 20)(15 26 17)(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 3)(5 16)(6 15)(7 14)(8 13)(9 25)(10 28)(11 27)(12 26)(17 36)(18 35)(19 34)(20 33)(21 31)(22 30)(23 29)(24 32)

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

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

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

He34D4 is a maximal subgroup of
He33SD16  He32D8  He33D8  He35SD16  He36SD16  He34D8  He36D8  He310SD16  C12⋊S3⋊S3  C12.84S32  C3⋊S3⋊D12  C12.86S32  C62.36D6  D4×C32⋊C6  (Q8×He3)⋊C2
He34D4 is a maximal quotient of
He34Q16  He36SD16  He34D8  C62.20D6  C62.21D6

31 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 4 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 12A 12B 12C ··· 12J order 1 2 2 2 3 3 3 3 3 3 4 6 6 6 6 6 6 6 6 6 6 12 12 12 ··· 12 size 1 1 18 18 2 3 3 6 6 6 2 2 3 3 6 6 6 18 18 18 18 2 2 6 ··· 6

31 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 6 6 6 type + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D4 D6 C3×S3 D12 C3×D4 S3×C6 C3×D12 C32⋊C6 C2×C32⋊C6 He3⋊4D4 kernel He3⋊4D4 C4×He3 C2×C32⋊C6 C12⋊S3 C3×C12 C2×C3⋊S3 C3×C12 He3 C3×C6 C12 C32 C32 C6 C3 C4 C2 C1 # reps 1 1 2 2 2 4 1 1 1 2 2 2 2 4 1 1 2

Matrix representation of He34D4 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
,
 3 7 0 0 0 0 6 10 0 0 0 0 0 0 3 7 0 0 0 0 6 10 0 0 0 0 0 0 3 7 0 0 0 0 6 10
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12 0 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],[3,6,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,7,10],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12,0,0,0] >;

He34D4 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_4D_4
% in TeX

G:=Group("He3:4D4");
// GroupNames label

G:=SmallGroup(216,51);
// by ID

G=gap.SmallGroup(216,51);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,79,1444,736,5189]);
// 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,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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