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G = He3⋊6D4  order 216 = 23·33

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

metabelian, supersoluble, monomial

Aliases: He3⋊6D4, C62⋊2S3, C62⋊2C6, C3⋊Dic3⋊C6, C32⋊7D4⋊C3, C6.18(S3×C6), (C3×C6).13D6, C32⋊3(C3×D4), C32⋊C12⋊4C2, C32⋊4(C3⋊D4), (C22×He3)⋊2C2, C22⋊3(C32⋊C6), (C2×He3).10C22, (C2×C3⋊S3)⋊2C6, (C3×C6).5(C2×C6), C3.2(C3×C3⋊D4), (C2×C32⋊C6)⋊4C2, (C2×C6).13(C3×S3), C2.5(C2×C32⋊C6), SmallGroup(216,60)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — He3⋊6D4
Chief series
C1 — C3 — C32 — C3×C6 — C2×He3 — C2×C32⋊C6 — He3⋊6D4
Lower central
C32 — C3×C6 — He3⋊6D4
Upper central
C1 — C2 — C22

Generators and relations for He3⋊6D4
 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, dbd-1=ebe=b-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 270 in 66 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, D4, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊D4, C3×D4, He3, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, C62, C32⋊C6, C2×He3, C2×He3, C3×C3⋊D4, C32⋊7D4, C32⋊C12, C2×C32⋊C6, C22×He3, He3⋊6D4
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, C3×S3, C3⋊D4, C3×D4, S3×C6, C32⋊C6, C3×C3⋊D4, C2×C32⋊C6, He3⋊6D4

Smallest permutation representation of He3⋊6D4
►On 36 points
Generators in S36
(1 16 11)(2 12 13)(3 14 9)(4 10 15)(5 31 26)(6 27 32)(7 29 28)(8 25 30)(17 36 22)(18 23 33)(19 34 24)(20 21 35)

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

(9 18 32)(10 19 29)(11 20 30)(12 17 31)(13 26 36)(14 27 33)(15 28 34)(16 25 35)

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

He3⋊6D4 is a maximal subgroup of   C62.8D6  C62.9D6  C62⋊D6  C62⋊2D6  C62.36D6  D4×C32⋊C6  C62.13D6
He3⋊6D4 is a maximal quotient of   C62.19D6  C62.21D6  He3⋊8SD16  He3⋊6D8  He3⋊6Q16  He3⋊10SD16  C62⋊3C12

31 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F 4 6A6B6C6D6E6F···6P6Q6R12A12B
order12223333334666666···6661212
size1121823366618222336···618181818

31 irreducible representations

dim1111111122222222666
type+++++++++
imageC1C2C2C2C3C6C6C6S3D4D6C3×S3C3⋊D4C3×D4S3×C6C3×C3⋊D4C32⋊C6C2×C32⋊C6He3⋊6D4
kernelHe3⋊6D4C32⋊C12C2×C32⋊C6C22×He3C32⋊7D4C3⋊Dic3C2×C3⋊S3C62C62He3C3×C6C2×C6C32C32C6C3C22C2C1
# reps1111222211122224112

Matrix representation of He3⋊6D4 ►in GL8(𝔽13)

10000000
01000000
00001000
00000100
00000010
00000001
00100000
00010000
,
10000000
01000000
001210000
001200000
000012100
000012000
000000121
000000120
,
30000000
03000000
00100000
00010000
000001200
000011200
000000121
000000120
,
64000000
77000000
000120000
001200000
000000012
000000120
000001200
000012000
,
10000000
1012000000
00010000
00100000
00000001
00000010
00000100
00001000

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

He3⋊6D4 in GAP, Magma, Sage, TeX 

{\rm He}_3\rtimes_6D_4
 
% in TeX
 
G:=Group("He3:6D4");
 
// GroupNames label
 
G:=SmallGroup(216,60);
 
// by ID
 
G=gap.SmallGroup(216,60);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,1444,736,5189]);
 
// 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,d*a*d^-1=e*a*e=a^-1,b*c=c*b,d*b*d^-1=e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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