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G = Dic3×S4order 288 = 25·32

Direct product of Dic3 and S4

direct product, non-abelian, soluble, monomial

Aliases: Dic3×S4, (C3×S4)⋊C4, C33(C4×S4), (C2×S4).S3, (C6×S4).C2, C2.1(S3×S4), C23.2S32, C6.10(C2×S4), (C2×A4).2D6, C6.7S41C2, (Dic3×A4)⋊2C2, A41(C2×Dic3), (C22×C6).2D6, (C6×A4).2C22, C221(S3×Dic3), (C22×Dic3)⋊1S3, (C2×C6)⋊1(C4×S3), (C3×A4)⋊1(C2×C4), SmallGroup(288,853)

Series: Derived Chief Lower central Upper central

C1C22C3×A4 — Dic3×S4
C1C22C2×C6C3×A4C6×A4Dic3×A4 — Dic3×S4
C3×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, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 542 in 118 conjugacy classes, 23 normal (19 characteristic)
C1, C2, C2 [×4], C3, C3 [×2], C4 [×6], C22, C22 [×6], S3 [×2], C6, C6 [×6], C2×C4 [×7], D4 [×4], C23, C23, C32, Dic3, Dic3 [×5], C12 [×3], A4, A4, D6, C2×C6, C2×C6 [×6], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3×S3 [×2], C3×C6, C4×S3, C2×Dic3 [×6], C2×C12, C3×D4 [×4], S4 [×2], C2×A4, C2×A4, C22×C6, C22×C6, C4×D4, C3×Dic3, C3⋊Dic3, C3×A4, S3×C6, C4×Dic3, C4⋊Dic3, C6.D4 [×2], A4⋊C4 [×2], C4×A4, C22×Dic3, C22×Dic3, C6×D4, C2×S4, S3×Dic3, C3×S4 [×2], C6×A4, D4×Dic3, C4×S4, C6.7S4, Dic3×A4, C6×S4, Dic3×S4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, Dic3 [×2], D6 [×2], C4×S3, C2×Dic3, S4, S32, C2×S4, S3×Dic3, C4×S4, S3×S4, Dic3×S4

Character table of Dic3×S4

 class 12A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H4I4J6A6B6C6D6E6F6G12A12B12C12D
 size 113366281633669918181818266812121612122424
ρ1111111111111111111111111111111    trivial
ρ21111-1-111111-1-111-1-1-1-11111-1-11-1-111    linear of order 2
ρ3111111111-1-111-1-1-1-1-1-1111111111-1-1    linear of order 2
ρ41111-1-1111-1-1-1-1-1-111111111-1-11-1-1-1-1    linear of order 2
ρ51-11-11-1111-ii-11i-i-ii-ii-11-1-1-11-11-1i-i    linear of order 4
ρ61-11-1-11111-ii1-1i-ii-ii-i-11-1-11-1-1-11i-i    linear of order 4
ρ71-11-11-1111i-i-11-iii-ii-i-11-1-1-11-11-1-ii    linear of order 4
ρ81-11-1-11111i-i1-1-ii-ii-ii-11-1-11-1-1-11-ii    linear of order 4
ρ92222002-1-1-2-200-2-20000222-100-10011    orthogonal lifted from D6
ρ102222-2-2-12-100-2-2000000-1-1-1211-11100    orthogonal lifted from D6
ρ11222222-12-10022000000-1-1-12-1-1-1-1-100    orthogonal lifted from S3
ρ122222002-1-12200220000222-100-100-1-1    orthogonal lifted from S3
ρ132-22-22-2-12-100-220000001-11-21-11-1100    symplectic lifted from Dic3, Schur index 2
ρ142-22-2-22-12-1002-20000001-11-2-1111-100    symplectic lifted from Dic3, Schur index 2
ρ152-22-2002-1-12i-2i00-2i2i0000-22-2100100i-i    complex lifted from C4×S3
ρ162-22-2002-1-1-2i2i002i-2i0000-22-2100100-ii    complex lifted from C4×S3
ρ1733-1-11130033-1-1-1-1-1-1113-1-10110-1-100    orthogonal lifted from S4
ρ1833-1-111300-3-3-1-11111-1-13-1-10110-1-100    orthogonal lifted from C2×S4
ρ1933-1-1-1-1300-3-31111-1-1113-1-10-1-101100    orthogonal lifted from C2×S4
ρ2033-1-1-1-13003311-1-111-1-13-1-10-1-101100    orthogonal lifted from S4
ρ213-3-111-13003i-3i1-1i-i-iii-i-3-110-110-1100    complex lifted from C4×S4
ρ223-3-111-1300-3i3i1-1-iii-i-ii-3-110-110-1100    complex lifted from C4×S4
ρ233-3-11-11300-3i3i-11-ii-iii-i-3-1101-101-100    complex lifted from C4×S4
ρ243-3-11-113003i-3i-11i-ii-i-ii-3-1101-101-100    complex lifted from C4×S4
ρ25444400-2-210000000000-2-2-2-20010000    orthogonal lifted from S32
ρ264-44-400-2-2100000000002-22200-10000    symplectic lifted from S3×Dic3, Schur index 2
ρ2766-2-222-30000-2-2000000-3110-1-101100    orthogonal lifted from S3×S4
ρ2866-2-2-2-2-3000022000000-3110110-1-100    orthogonal lifted from S3×S4
ρ296-6-222-2-300002-200000031-101-101-100    symplectic faithful, Schur index 2
ρ306-6-22-22-30000-2200000031-10-110-1100    symplectic faithful, Schur index 2

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

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

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

Matrix representation of Dic3×S4 in GL5(𝔽13)

40000
910000
00100
00010
00001
,
512000
08000
001200
000120
000012
,
10000
01000
00100
000120
000012
,
10000
01000
001200
000120
00001
,
10000
01000
000120
000012
00100
,
10000
01000
00010
00100
000012

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

Dic3×S4 in GAP, Magma, Sage, TeX

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

G:=Group("Dic3xS4");
// GroupNames label

G:=SmallGroup(288,853);
// by ID

G=gap.SmallGroup(288,853);
# by ID

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

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