Copied to
clipboard

G = Dic3.S32  order 432 = 24·33

4th non-split extension by Dic3 of S32 acting via S32/C3xS3=C2

metabelian, supersoluble, monomial

Aliases: Dic3.4S32, C32:2Q8:6S3, C6.D6:3S3, C33:7(C4oD4), C33:8D4:6C2, C33:9D4:10C2, C3:Dic3.33D6, C3:3(D6.D6), C3:1(D6.6D6), C32:9(C4oD12), (C3xDic3).22D6, C32:5(Q8:3S3), (C32xC6).19C23, (C32xDic3).7C22, C2.19S33, C6.19(C2xS32), C33:8(C2xC4):4C2, (C2xC3:S3).19D6, (C3xC6.D6):3C2, (C3xC32:2Q8):7C2, (C6xC3:S3).23C22, (C3xC6).68(C22xS3), (C3xC3:Dic3).16C22, (C2xC33:C2).6C22, SmallGroup(432,612)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32xC6 — Dic3.S32
Chief series
C1 — C3 — C32 — C33 — C32xC6 — C32xDic3 — C3xC6.D6 — Dic3.S32
Lower central
C33 — C32xC6 — Dic3.S32
Upper central
C1 — C2

Generators and relations for Dic3.S32
 G = < a,b,c,d,e | a6=c2=d3=e2=1, b6=a3, bab-1=cac=eae=a-1, ad=da, cbc=b5, bd=db, be=eb, cd=dc, ece=a3c, ede=d-1 >

Subgroups: 1476 in 218 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C2xC4, D4, Q8, C32, C32, C32, Dic3, Dic3, Dic3, C12, D6, C2xC6, C4oD4, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, Dic6, C4xS3, D12, C3:D4, C2xC12, C3xQ8, C33, C3xDic3, C3xDic3, C3:Dic3, C3xC12, S3xC6, C2xC3:S3, C2xC3:S3, C4oD12, Q8:3S3, C3xC3:S3, C33:C2, C32xC6, C6.D6, C6.D6, D6:S3, C3:D12, C32:2Q8, C3xDic6, S3xC12, C4xC3:S3, C12:S3, C32xDic3, C32xDic3, C3xC3:Dic3, C6xC3:S3, C2xC33:C2, D6.D6, D6.6D6, C3xC6.D6, C3xC32:2Q8, C33:8(C2xC4), C33:8D4, C33:9D4, Dic3.S32
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C22xS3, S32, C4oD12, Q8:3S3, C2xS32, D6.D6, D6.6D6, S33, Dic3.S32

Permutation representations of Dic3.S32
►On 24 points - transitive group 24T1300
Generators in S24
(1 11 9 7 5 3)(2 4 6 8 10 12)(13 23 21 19 17 15)(14 16 18 20 22 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)

(1 17)(2 22)(3 15)(4 20)(5 13)(6 18)(7 23)(8 16)(9 21)(10 14)(11 19)(12 24)

(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)

(1 24)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)
 
G:=sub<Sym(24)| (1,11,9,7,5,3)(2,4,6,8,10,12)(13,23,21,19,17,15)(14,16,18,20,22,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,17)(2,22)(3,15)(4,20)(5,13)(6,18)(7,23)(8,16)(9,21)(10,14)(11,19)(12,24), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,24)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)>;
 
G:=Group( (1,11,9,7,5,3)(2,4,6,8,10,12)(13,23,21,19,17,15)(14,16,18,20,22,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,17)(2,22)(3,15)(4,20)(5,13)(6,18)(7,23)(8,16)(9,21)(10,14)(11,19)(12,24), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,24)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23) );
 
G=PermutationGroup([[(1,11,9,7,5,3),(2,4,6,8,10,12),(13,23,21,19,17,15),(14,16,18,20,22,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,17),(2,22),(3,15),(4,20),(5,13),(6,18),(7,23),(8,16),(9,21),(10,14),(11,19),(12,24)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,24),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23)]])
 
G:=TransitiveGroup(24,1300);
 

45 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E3F3G4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J6K12A···12H12I···12P12Q
order122223333333444446666666666612···1212···1212
size1118185422244483366182224448181818186···612···1236

45 irreducible representations

dim11111122222224444488
type+++++++++++++++++
imageC1C2C2C2C2C2S3S3D6D6D6C4oD4C4oD12S32Q8:3S3C2xS32D6.D6D6.6D6S33Dic3.S32
kernelDic3.S32C3xC6.D6C3xC32:2Q8C33:8(C2xC4)C33:8D4C33:9D4C6.D6C32:2Q8C3xDic3C3:Dic3C2xC3:S3C33C32Dic3C32C6C3C3C2C1
# reps12112121612283132411

Matrix representation of Dic3.S32 ►in GL8(Z)

01000000
-11000000
00010000
00-110000
00000100
0000-1100
00000001
000000-11
,
00000010
0000001-1
00001000
00001-100
001-10000
000-10000
1-1000000
0-1000000
,
001-10000
000-10000
1-1000000
0-1000000
000000-10
000000-11
0000-1000
0000-1100
,
-11000000
-10000000
000-10000
001-10000
0000-1100
0000-1000
0000000-1
0000001-1
,
00000-100
0000-1000
00000001
00000010
0-1000000
-10000000
00010000
00100000

G:=sub<GL(8,Integers())| [0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0],[-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1],[0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0] >;
 

Dic3.S32 in GAP, Magma, Sage, TeX 

{\rm Dic}_3.S_3^2
 
% in TeX
 
G:=Group("Dic3.S3^2");
 
// GroupNames label
 
G:=SmallGroup(432,612);
 
// by ID
 
G=gap.SmallGroup(432,612);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,135,58,298,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=c^2=d^3=e^2=1,b^6=a^3,b*a*b^-1=c*a*c=e*a*e=a^-1,a*d=d*a,c*b*c=b^5,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^3*c,e*d*e=d^-1>;
 
// generators/relations
 

׿
x
:
Z
F
o
wr
Q
<