Copied to
clipboard

G = Dic3⋊S4order 288 = 25·32

The semidirect product of Dic3 and S4 acting via S4/A4=C2

non-abelian, soluble, monomial

Aliases: Dic3⋊S4, (C2×C6)⋊D12, C32(C4⋊S4), (C2×S4)⋊1S3, (C6×S4)⋊2C2, (C3×A4)⋊1D4, C23.4S32, C6.12(C2×S4), C2.13(S3×S4), (C2×A4).4D6, A41(C3⋊D4), (Dic3×A4)⋊1C2, (C22×C6).4D6, (C6×A4).4C22, (C22×Dic3)⋊3S3, C221(C3⋊D12), (C2×C3⋊S4)⋊1C2, SmallGroup(288,855)

Series: Derived Chief Lower central Upper central

C1C22C6×A4 — Dic3⋊S4
C1C22C2×C6C3×A4C6×A4Dic3×A4 — Dic3⋊S4
C3×A4C6×A4 — Dic3⋊S4
C1C2

Generators and relations for Dic3⋊S4
 G = < a,b,c,d,e,f | a6=c2=d2=e3=f2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a3b, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 786 in 122 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2 [×4], C3, C3 [×2], C4 [×4], C22, C22 [×8], S3 [×5], C6, C6 [×5], C2×C4 [×4], D4 [×6], C23, C23 [×2], C32, Dic3, Dic3 [×2], C12 [×2], A4, A4, D6 [×6], C2×C6, C2×C6 [×5], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3, C3⋊S3, C3×C6, D12, C2×Dic3 [×3], C3⋊D4 [×4], C2×C12, C3×D4 [×2], S4 [×4], C2×A4, C2×A4, C22×S3, C22×C6, C22×C6, C4⋊D4, C3×Dic3, C3×A4, S3×C6, C2×C3⋊S3, Dic3⋊C4, D6⋊C4, C6.D4, C4×A4, C22×Dic3, C2×C3⋊D4 [×2], C6×D4, C2×S4, C2×S4 [×2], C3⋊D12, C3×S4, C3⋊S4, C6×A4, C23.14D6, C4⋊S4, Dic3×A4, C6×S4, C2×C3⋊S4, Dic3⋊S4
Quotients: C1, C2 [×3], C22, S3 [×2], D4, D6 [×2], D12, C3⋊D4, S4, S32, C2×S4, C3⋊D12, C4⋊S4, S3×S4, Dic3⋊S4

Character table of Dic3⋊S4

 class 12A2B2C2D2E3A3B3C4A4B4C4D6A6B6C6D6E6F6G12A12B12C12D
 size 1133123628166121836266812121612122424
ρ1111111111111111111111111    trivial
ρ21111-11111-1-1-111111-1-11-1-1-1-1    linear of order 2
ρ311111-1111-11-1-1111111111-1-1    linear of order 2
ρ41111-1-11111-11-11111-1-11-1-111    linear of order 2
ρ52222-20-12-10-200-1-1-1211-11100    orthogonal lifted from D6
ρ62-22-2002220000-22-2-200-20000    orthogonal lifted from D4
ρ72222002-1-12020222-100-100-1-1    orthogonal lifted from S3
ρ82222002-1-1-20-20222-100-10011    orthogonal lifted from D6
ρ9222220-12-10200-1-1-12-1-1-1-1-100    orthogonal lifted from S3
ρ102-22-2002-1-10000-22-2100100-33    orthogonal lifted from D12
ρ112-22-2002-1-10000-22-21001003-3    orthogonal lifted from D12
ρ122-22-200-12-100001-11-2-3--31-3--300    complex lifted from C3⋊D4
ρ132-22-200-12-100001-11-2--3-31--3-300    complex lifted from C3⋊D4
ρ1433-1-11-1300-3-1113-1-10110-1-100    orthogonal lifted from C2×S4
ρ1533-1-1113003-1-1-13-1-10110-1-100    orthogonal lifted from S4
ρ1633-1-1-11300-311-13-1-10-1-101100    orthogonal lifted from C2×S4
ρ1733-1-1-1-130031-113-1-10-1-101100    orthogonal lifted from S4
ρ184-44-400-2-2100002-22200-10000    orthogonal lifted from C3⋊D12
ρ19444400-2-210000-2-2-2-20010000    orthogonal lifted from S32
ρ2066-2-220-3000-200-3110-1-101100    orthogonal lifted from S3×S4
ρ2166-2-2-20-3000200-3110110-1-100    orthogonal lifted from S3×S4
ρ226-6-22006000000-6-2200000000    orthogonal lifted from C4⋊S4
ρ236-6-2200-300000031-10--3-30-3--300    complex faithful
ρ246-6-2200-300000031-10-3--30--3-300    complex faithful

Smallest permutation representation of Dic3⋊S4
On 36 points
Generators in S36
(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 23 4 20)(2 22 5 19)(3 21 6 24)(7 33 10 36)(8 32 11 35)(9 31 12 34)(13 27 16 30)(14 26 17 29)(15 25 18 28)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(19 22)(20 23)(21 24)(31 34)(32 35)(33 36)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)
(1 11 17)(2 12 18)(3 7 13)(4 8 14)(5 9 15)(6 10 16)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)
(1 23)(2 24)(3 19)(4 20)(5 21)(6 22)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)

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

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

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

Matrix representation of Dic3⋊S4 in GL5(𝔽13)

710000
107000
00100
00010
00001
,
37000
610000
001200
000120
000012
,
10000
01000
00010
00100
00121212
,
10000
01000
00121212
00001
00010
,
10000
01000
00100
00001
00121212
,
012000
120000
00100
00001
00010

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

Dic3⋊S4 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes S_4
% in TeX

G:=Group("Dic3:S4");
// GroupNames label

G:=SmallGroup(288,855);
// by ID

G=gap.SmallGroup(288,855);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,28,85,234,1684,3036,782,1777,1350]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^6=c^2=d^2=e^3=f^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a^3*b,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations

Export

Character table of Dic3⋊S4 in TeX

׿
×
𝔽