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G = He3:3D4  order 216 = 23·33

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

non-abelian, supersoluble, monomial

Aliases: He3:3D4, C32:D12, C6.20S32, (C3xC6).5D6, C3:Dic3:3S3, C32:C12:1C2, C32:2(C3:D4), C2.5(C32:D6), C3.3(C3:D12), (C2xHe3).5C22, (C2xC3:S3):2S3, (C2xC32:C6):2C2, (C2xHe3:C2):1C2, SmallGroup(216,37)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C2xHe3 — He3:3D4
Chief series
C1 — C3 — C32 — He3 — C2xHe3 — C2xC32:C6 — He3:3D4
Lower central
He3 — C2xHe3 — He3:3D4
Upper central
C1 — C2

Generators and relations for He3:3D4
 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, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 368 in 66 conjugacy classes, 16 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, D4, C32, C32, Dic3, C12, D6, C2xC6, C3xS3, C3:S3, C3xC6, C3xC6, D12, C3:D4, He3, C3xDic3, C3:Dic3, S3xC6, C2xC3:S3, C32:C6, He3:C2, C2xHe3, D6:S3, C3:D12, C32:C12, C2xC32:C6, C2xHe3:C2, He3:3D4
Quotients: C1, C2, C22, S3, D4, D6, D12, C3:D4, S32, C3:D12, C32:D6, He3:3D4

Character table of He3:3D4

 class 12A2B2C3A3B3C3D46A6B6C6D6E6F6G6H12A12B
 size 111818266121826612181818181818
ρ11111111111111111111    trivial
ρ211-1-1111111111-1-1-1-111    linear of order 2
ρ311-111111-11111-111-1-1-1    linear of order 2
ρ4111-11111-111111-1-11-1-1    linear of order 2
ρ52-20022220-2-2-2-2000000    orthogonal lifted from D4
ρ622022-12-102-12-10-1-1000    orthogonal lifted from S3
ρ7220022-1-1-222-1-1000011    orthogonal lifted from D6
ρ8220022-1-1222-1-10000-1-1    orthogonal lifted from S3
ρ9220-22-12-102-12-1011000    orthogonal lifted from D6
ρ102-20022-1-10-2-2110000√3-√3    orthogonal lifted from D12
ρ112-20022-1-10-2-2110000-√3√3    orthogonal lifted from D12
ρ122-2002-12-10-21-210-√-3√-3000    complex lifted from C3:D4
ρ132-2002-12-10-21-210√-3-√-3000    complex lifted from C3:D4
ρ144-4004-2-210-422-1000000    orthogonal lifted from C3:D12
ρ1544004-2-2104-2-21000000    orthogonal lifted from S32
ρ166620-30000-3000-100-100    orthogonal lifted from C32:D6
ρ1766-20-30000-3000100100    orthogonal lifted from C32:D6
ρ186-600-300003000-√-300√-300    complex faithful
ρ196-600-300003000√-300-√-300    complex faithful

Smallest permutation representation of He3:3D4
►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 30 22)(3 23 31)(4 32 24)(5 35 11)(6 12 36)(7 33 9)(8 10 34)(13 19 28)(14 25 20)(15 17 26)(16 27 18)

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

He3:3D4 is a maximal subgroup of   C12.84S32  C12.91S32  C12.S32  C3:S3:D12  C62.8D6  C62:D6  C62:2D6
He3:3D4 is a maximal quotient of   He3:3D8  He3:4SD16  He3:5SD16  He3:3Q16  C62.D6  C62.4D6  C62.5D6

Matrix representation of He3:3D4 ►in GL6(F13)

001000
000100
000010
000001
100000
010000
,
1210000
1200000
0012100
0012000
0000121
0000120
,
100000
010000
0001200
0011200
0000121
0000120
,
4110000
290000
0041100
002900
0000411
000029
,
290000
4110000
000029
0000411
002900
0041100

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],[4,2,0,0,0,0,11,9,0,0,0,0,0,0,4,2,0,0,0,0,11,9,0,0,0,0,0,0,4,2,0,0,0,0,11,9],[2,4,0,0,0,0,9,11,0,0,0,0,0,0,0,0,2,4,0,0,0,0,9,11,0,0,2,4,0,0,0,0,9,11,0,0] >;
 

He3:3D4 in GAP, Magma, Sage, TeX 

{\rm He}_3\rtimes_3D_4
 
% in TeX
 
G:=Group("He3:3D4");
 
// GroupNames label
 
G:=SmallGroup(216,37);
 
// by ID
 
G=gap.SmallGroup(216,37);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,201,1444,382,5189,2603]);
 
// 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,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

Export

Character table of He3:3D4 in TeX

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