Copied to
clipboard

G = Dic3.S32order 432 = 24·33

4th non-split extension by Dic3 of S32 acting via S32/C3×S3=C2

metabelian, supersoluble, monomial

Aliases: Dic3.4S32, C322Q86S3, C6.D63S3, C337(C4○D4), C338D46C2, C339D410C2, C3⋊Dic3.33D6, C33(D6.D6), C31(D6.6D6), C329(C4○D12), (C3×Dic3).22D6, C325(Q83S3), (C32×C6).19C23, (C32×Dic3).7C22, C2.19S33, C6.19(C2×S32), C338(C2×C4)⋊4C2, (C2×C3⋊S3).19D6, (C3×C6.D6)⋊3C2, (C3×C322Q8)⋊7C2, (C6×C3⋊S3).23C22, (C3×C6).68(C22×S3), (C3×C3⋊Dic3).16C22, (C2×C33⋊C2).6C22, SmallGroup(432,612)

Series: Derived Chief Lower central Upper central

C1C32×C6 — Dic3.S32
C1C3C32C33C32×C6C32×Dic3C3×C6.D6 — Dic3.S32
C33C32×C6 — Dic3.S32
C1C2

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 [×3], C3, C3 [×2], C3 [×4], C4 [×4], C22 [×3], S3 [×15], C6, C6 [×2], C6 [×6], C2×C4 [×3], D4 [×3], Q8, C32, C32 [×2], C32 [×4], Dic3, Dic3 [×2], Dic3 [×3], C12 [×10], D6 [×13], C2×C6 [×2], C4○D4, C3×S3 [×6], C3⋊S3 [×11], C3×C6, C3×C6 [×2], C3×C6 [×4], Dic6 [×2], C4×S3 [×8], D12 [×7], C3⋊D4 [×4], C2×C12 [×2], C3×Q8, C33, C3×Dic3 [×6], C3×Dic3 [×6], C3⋊Dic3, C3×C12 [×3], S3×C6 [×6], C2×C3⋊S3 [×2], C2×C3⋊S3 [×7], C4○D12 [×2], Q83S3, C3×C3⋊S3 [×2], C33⋊C2, C32×C6, C6.D6 [×2], C6.D6 [×3], D6⋊S3, C3⋊D12 [×8], C322Q8, C3×Dic6 [×2], S3×C12 [×4], C4×C3⋊S3, C12⋊S3 [×2], C32×Dic3, C32×Dic3 [×2], C3×C3⋊Dic3, C6×C3⋊S3 [×2], C2×C33⋊C2, D6.D6, D6.6D6 [×2], C3×C6.D6 [×2], C3×C322Q8, C338(C2×C4), C338D4 [×2], C339D4, Dic3.S32
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], C23, D6 [×9], C4○D4, C22×S3 [×3], S32 [×3], C4○D12 [×2], Q83S3, C2×S32 [×3], D6.D6, D6.6D6 [×2], 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 19)(2 24)(3 17)(4 22)(5 15)(6 20)(7 13)(8 18)(9 23)(10 16)(11 21)(12 14)
(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 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 13)

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

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

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

G:=TransitiveGroup(24,1300);

45 conjugacy classes

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

45 irreducible representations

dim11111122222224444488
type+++++++++++++++++
imageC1C2C2C2C2C2S3S3D6D6D6C4○D4C4○D12S32Q83S3C2×S32D6.D6D6.6D6S33Dic3.S32
kernelDic3.S32C3×C6.D6C3×C322Q8C338(C2×C4)C338D4C339D4C6.D6C322Q8C3×Dic3C3⋊Dic3C2×C3⋊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

׿
×
𝔽