Copied to
clipboard

G = He32D4order 216 = 23·33

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

non-abelian, supersoluble, monomial

Aliases: He32D4, C6.18S32, (C3×C6).3D6, He33C41C2, C321(C3⋊D4), C2.4(C32⋊D6), C3.1(D6⋊S3), (C2×He3).3C22, (C2×C3⋊S3)⋊1S3, (C2×C32⋊C6)⋊1C2, SmallGroup(216,35)

Series: Derived Chief Lower central Upper central

C1C3C2×He3 — He32D4
C1C3C32He3C2×He3C2×C32⋊C6 — He32D4
He3C2×He3 — He32D4
C1C2

Generators and relations for He32D4
 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, ebe=b-1, dcd-1=c-1, ce=ec, ede=d-1 >

Subgroups: 352 in 62 conjugacy classes, 16 normal (8 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, D4, C32, C32, Dic3, C12, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, D12, C3⋊D4, He3, C3×Dic3, S3×C6, C2×C3⋊S3, C32⋊C6, C2×He3, C3⋊D12, He33C4, C2×C32⋊C6, He32D4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊D4, S32, D6⋊S3, C32⋊D6, He32D4

Character table of He32D4

 class 12A2B2C3A3B3C3D46A6B6C6D6E6F6G6H12A12B
 size 111818266121826612181818181818
ρ11111111111111111111    trivial
ρ211-1-1111111111-1-1-1-111    linear of order 2
ρ3111-11111-11111-111-1-1-1    linear of order 2
ρ411-111111-111111-1-11-1-1    linear of order 2
ρ52-20022220-2-2-2-2000000    orthogonal lifted from D4
ρ6222022-1-1022-1-10-1-1000    orthogonal lifted from S3
ρ722022-12-102-12-1-100-100    orthogonal lifted from S3
ρ822-2022-1-1022-1-1011000    orthogonal lifted from D6
ρ9220-22-12-102-12-1100100    orthogonal lifted from D6
ρ102-20022-1-10-2-2110-3--3000    complex lifted from C3⋊D4
ρ112-2002-12-10-21-21--300-300    complex lifted from C3⋊D4
ρ122-20022-1-10-2-2110--3-3000    complex lifted from C3⋊D4
ρ132-2002-12-10-21-21-300--300    complex lifted from C3⋊D4
ρ1444004-2-2104-2-21000000    orthogonal lifted from S32
ρ154-4004-2-210-422-1000000    symplectic lifted from D6⋊S3, Schur index 2
ρ166600-3000-2-3000000011    orthogonal lifted from C32⋊D6
ρ176600-30002-30000000-1-1    orthogonal lifted from C32⋊D6
ρ186-600-30000300000003-3    orthogonal faithful
ρ196-600-3000030000000-33    orthogonal faithful

Smallest permutation representation of He32D4
On 36 points
Generators in S36
(1 30 13)(2 14 31)(3 32 15)(4 16 29)(5 18 9)(6 10 19)(7 20 11)(8 12 17)(21 34 28)(22 25 35)(23 36 26)(24 27 33)
(1 5 24)(2 6 21)(3 7 22)(4 8 23)(9 33 13)(10 34 14)(11 35 15)(12 36 16)(17 26 29)(18 27 30)(19 28 31)(20 25 32)
(9 13 33)(10 34 14)(11 15 35)(12 36 16)(17 29 26)(18 27 30)(19 31 28)(20 25 32)
(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 23)(6 22)(7 21)(8 24)(9 36)(10 35)(11 34)(12 33)(13 16)(14 15)(17 27)(18 26)(19 25)(20 28)(29 30)(31 32)

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

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

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

He32D4 is a maximal subgroup of   He32SD16  He3⋊D8  C12⋊S3⋊S3  C12.91S32  C12.86S32  C62.9D6  C62⋊D6
He32D4 is a maximal quotient of   He33SD16  He32D8  He32Q16  C62.3D6  C62.4D6

Matrix representation of He32D4 in GL6(𝔽13)

0000121
33001112
000040
100040
0010100
0001100
,
0120000
1120000
90121200
041000
30001212
0100010
,
100000
010000
99121200
001000
000001
33001212
,
370000
6100000
10800107
530063
10510700
836300
,
730000
1060000
350036
12100310
383600
4931000

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

He32D4 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes_2D_4
% in TeX

G:=Group("He3:2D4");
// GroupNames label

G:=SmallGroup(216,35);
// by ID

G=gap.SmallGroup(216,35);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,201,1444,382,5189,2603]);
// 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,d*a*d^-1=e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^-1>;
// generators/relations

Export

Character table of He32D4 in TeX

׿
×
𝔽