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## G = He3⋊7D4order 216 = 23·33

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — He3⋊7D4
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×He3⋊C2 — He3⋊7D4
 Lower central He3 — C2×He3 — He3⋊7D4
 Upper central C1 — C6 — C2×C6

Generators and relations for He37D4
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, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 310 in 88 conjugacy classes, 24 normal (12 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4, C22, C22, S3 [×4], C6, C6 [×10], D4, C32 [×4], Dic3 [×4], C12, D6 [×4], C2×C6, C2×C6 [×5], C3×S3 [×4], C3×C6 [×4], C3×C6 [×4], C3⋊D4 [×4], C3×D4, He3, C3×Dic3 [×4], S3×C6 [×4], C62 [×4], He3⋊C2, C2×He3, C2×He3, C3×C3⋊D4 [×4], He33C4, C2×He3⋊C2, C22×He3, He37D4
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], C3⋊S3, C3⋊D4 [×4], C2×C3⋊S3, He3⋊C2, C327D4, C2×He3⋊C2, He37D4

Smallest permutation representation of He37D4
On 36 points
Generators in S36
(1 35 22)(2 23 36)(3 33 24)(4 21 34)(5 11 16)(6 13 12)(7 9 14)(8 15 10)(17 26 31)(18 32 27)(19 28 29)(20 30 25)
(1 30 5)(2 31 6)(3 32 7)(4 29 8)(9 33 27)(10 34 28)(11 35 25)(12 36 26)(13 23 17)(14 24 18)(15 21 19)(16 22 20)
(9 33 27)(10 28 34)(11 35 25)(12 26 36)(13 23 17)(14 18 24)(15 21 19)(16 20 22)
(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)(6 8)(9 14)(10 13)(11 16)(12 15)(17 28)(18 27)(19 26)(20 25)(21 36)(22 35)(23 34)(24 33)(29 31)

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

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

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

He37D4 is a maximal subgroup of   C62.8D6  C62⋊D6  C62.47D6  D4×He3⋊C2  C62.16D6
He37D4 is a maximal quotient of   C62.29D6  C62.31D6  He37D8  He39SD16  He311SD16  He37Q16  C624Dic3

31 conjugacy classes

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

31 irreducible representations

 dim 1 1 1 1 2 2 2 2 3 3 6 type + + + + + + + image C1 C2 C2 C2 S3 D4 D6 C3⋊D4 He3⋊C2 C2×He3⋊C2 He3⋊7D4 kernel He3⋊7D4 He3⋊3C4 C2×He3⋊C2 C22×He3 C62 He3 C3×C6 C32 C22 C2 C1 # reps 1 1 1 1 4 1 4 8 4 4 2

Matrix representation of He37D4 in GL5(𝔽13)

 0 1 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 12 12 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 9
,
 11 2 0 0 0 4 2 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0
,
 1 1 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(13))| [0,12,0,0,0,1,12,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[12,1,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,9],[11,4,0,0,0,2,2,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,1,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

He37D4 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_7D_4
% in TeX

G:=Group("He3:7D4");
// GroupNames label

G:=SmallGroup(216,72);
// by ID

G=gap.SmallGroup(216,72);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,387,1444,382]);
// 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,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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