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## G = He3⋊5D4order 216 = 23·33

### 2nd semidirect product of He3 and D4 acting via D4/C4=C2

Aliases: He35D4, C324D12, (C3×C12)⋊2S3, C4⋊(He3⋊C2), (C4×He3)⋊2C2, (C3×C6).18D6, C12.8(C3⋊S3), C3.2(C12⋊S3), (C2×He3).13C22, C6.28(C2×C3⋊S3), (C2×He3⋊C2)⋊2C2, C2.4(C2×He3⋊C2), SmallGroup(216,68)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — He3⋊5D4
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C2×He3⋊C2 — He3⋊5D4
 Lower central He3 — C2×He3 — He3⋊5D4
 Upper central C1 — C6 — C12

Generators and relations for He35D4
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, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 358 in 88 conjugacy classes, 24 normal (10 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4, C22 [×2], S3 [×8], C6, C6 [×6], D4, C32 [×4], C12, C12 [×4], D6 [×8], C2×C6 [×2], C3×S3 [×8], C3×C6 [×4], D12 [×4], C3×D4, He3, C3×C12 [×4], S3×C6 [×8], He3⋊C2 [×2], C2×He3, C3×D12 [×4], C4×He3, C2×He3⋊C2 [×2], He35D4
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], C3⋊S3, D12 [×4], C2×C3⋊S3, He3⋊C2, C12⋊S3, C2×He3⋊C2, He35D4

Smallest permutation representation of He35D4
On 36 points
Generators in S36
(1 16 24)(2 13 21)(3 14 22)(4 15 23)(5 25 35)(6 26 36)(7 27 33)(8 28 34)(9 18 29)(10 19 30)(11 20 31)(12 17 32)
(1 7 29)(2 8 30)(3 5 31)(4 6 32)(9 16 27)(10 13 28)(11 14 25)(12 15 26)(17 23 36)(18 24 33)(19 21 34)(20 22 35)
(1 33 27)(2 34 28)(3 35 25)(4 36 26)(5 20 11)(6 17 12)(7 18 9)(8 19 10)(13 30 21)(14 31 22)(15 32 23)(16 29 24)
(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 8)(6 7)(9 17)(10 20)(11 19)(12 18)(13 22)(14 21)(15 24)(16 23)(25 34)(26 33)(27 36)(28 35)(29 32)(30 31)

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

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

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

He35D4 is a maximal subgroup of
He33D8  He34SD16  He37SD16  He35D8  He37D8  He311SD16  C12.S32  C3⋊S3⋊D12  C62.47D6  D4×He3⋊C2  He35D4⋊C2
He35D4 is a maximal quotient of
He37SD16  He35D8  He35Q16  C62.30D6  C62.31D6

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 1 1 6 6 6 6 2 1 1 6 6 6 6 18 18 18 18 2 2 6 ··· 6

31 irreducible representations

 dim 1 1 1 2 2 2 2 3 3 6 type + + + + + + + image C1 C2 C2 S3 D4 D6 D12 He3⋊C2 C2×He3⋊C2 He3⋊5D4 kernel He3⋊5D4 C4×He3 C2×He3⋊C2 C3×C12 He3 C3×C6 C32 C4 C2 C1 # reps 1 1 2 4 1 4 8 4 4 2

Matrix representation of He35D4 in GL5(𝔽13)

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

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

He35D4 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_5D_4
% in TeX

G:=Group("He3:5D4");
// GroupNames label

G:=SmallGroup(216,68);
// by ID

G=gap.SmallGroup(216,68);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,387,1444,382]);
// 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,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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