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G = He36D4order 216 = 23·33

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

metabelian, supersoluble, monomial

Aliases: He36D4, C622S3, C622C6, C3⋊Dic3⋊C6, C327D4⋊C3, C6.18(S3×C6), (C3×C6).13D6, C323(C3×D4), C32⋊C124C2, C324(C3⋊D4), (C22×He3)⋊2C2, C223(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

C1C3×C6 — He36D4
C1C3C32C3×C6C2×He3C2×C32⋊C6 — He36D4
C32C3×C6 — He36D4
C1C2C22

Generators and relations for He36D4
 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 [×2], C3, C3 [×3], C4, C22, C22, S3 [×2], C6, C6 [×9], D4, C32 [×2], C32, Dic3 [×2], C12, D6 [×2], C2×C6, C2×C6 [×4], C3×S3, C3⋊S3, C3×C6 [×2], C3×C6 [×5], C3⋊D4 [×2], C3×D4, He3, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62 [×2], C62, C32⋊C6, C2×He3, C2×He3, C3×C3⋊D4, C327D4, C32⋊C12, C2×C32⋊C6, C22×He3, He36D4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, C3×S3, C3⋊D4, C3×D4, S3×C6, C32⋊C6, C3×C3⋊D4, C2×C32⋊C6, He36D4

Smallest permutation representation of He36D4
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)])

He36D4 is a maximal subgroup of   C62.8D6  C62.9D6  C62⋊D6  C622D6  C62.36D6  D4×C32⋊C6  C62.13D6
He36D4 is a maximal quotient of   C62.19D6  C62.21D6  He38SD16  He36D8  He36Q16  He310SD16  C623C12

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⋊C6He36D4
kernelHe36D4C32⋊C12C2×C32⋊C6C22×He3C327D4C3⋊Dic3C2×C3⋊S3C62C62He3C3×C6C2×C6C32C32C6C3C22C2C1
# reps1111222211122224112

Matrix representation of He36D4 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] >;

He36D4 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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