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G = Dic3:2S4order 288 = 25·32

The semidirect product of Dic3 and S4 acting through Inn(Dic3)

non-abelian, soluble, monomial

Aliases: Dic3:2S4, C3:S4:C4, C3:1(C4xS4), A4:C4:2S3, A4:1(C4xS3), C2.2(S3xS4), C23.3S32, C6.11(C2xS4), (C2xA4).3D6, (Dic3xA4):3C2, (C22xC6).3D6, (C6xA4).3C22, C22:(C6.D6), (C22xDic3):2S3, (C2xC3:S4).C2, (C2xC6):2(C4xS3), (C3xA4:C4):2C2, (C3xA4):2(C2xC4), SmallGroup(288,854)

Series: Derived Chief Lower central Upper central

C1C22C3xA4 — Dic3:2S4
C1C22C2xC6C3xA4C6xA4Dic3xA4 — Dic3:2S4
C3xA4 — Dic3:2S4
C1C2

Generators and relations for Dic3:2S4
 G = < a,b,c,d,e,f | a6=c2=d2=e3=f2=1, b2=a3, bab-1=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 694 in 122 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C23, C32, Dic3, Dic3, C12, A4, A4, D6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C22xC4, C2xD4, C3:S3, C3xC6, C4xS3, C2xDic3, C3:D4, C2xC12, S4, C2xA4, C2xA4, C22xS3, C22xC6, C4xD4, C3xDic3, C3xA4, C2xC3:S3, C4xDic3, Dic3:C4, D6:C4, C3xC22:C4, A4:C4, C4xA4, S3xC2xC4, C22xDic3, C2xC3:D4, C2xS4, C6.D6, C3:S4, C6xA4, Dic3:4D4, C4xS4, C3xA4:C4, Dic3xA4, C2xC3:S4, Dic3:2S4
Quotients: C1, C2, C4, C22, S3, C2xC4, D6, C4xS3, S4, S32, C2xS4, C6.D6, C4xS4, S3xS4, Dic3:2S4

Character table of Dic3:2S4

 class 12A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H4I4J6A6B6C6D6E12A12B12C12D12E12F
 size 113318182816336666991818266816121212122424
ρ1111111111111111111111111111111    trivial
ρ21111-1-1111-1-11111-1-1-1-1111111111-1-1    linear of order 2
ρ3111111111-1-1-1-1-1-1-1-11111111-1-1-1-1-1-1    linear of order 2
ρ41111-1-111111-1-1-1-111-1-111111-1-1-1-111    linear of order 2
ρ51-11-1-11111i-i-iii-ii-i-11-11-1-1-1i-i-iii-i    linear of order 4
ρ61-11-11-1111-ii-iii-i-ii1-1-11-1-1-1i-i-ii-ii    linear of order 4
ρ71-11-1-11111-iii-i-ii-ii-11-11-1-1-1-iii-i-ii    linear of order 4
ρ81-11-11-1111i-ii-i-iii-i1-1-11-1-1-1-iii-ii-i    linear of order 4
ρ9222200-12-100-2-2-2-20000-1-1-12-1111100    orthogonal lifted from D6
ρ102222002-1-1-2-20000-2-200222-1-1000011    orthogonal lifted from D6
ρ11222200-12-10022220000-1-1-12-1-1-1-1-100    orthogonal lifted from S3
ρ122222002-1-12200002200222-1-10000-1-1    orthogonal lifted from S3
ρ132-22-2002-1-1-2i2i0000-2i2i00-22-2110000i-i    complex lifted from C4xS3
ρ142-22-200-12-100-2i2i2i-2i00001-11-21-iii-i00    complex lifted from C4xS3
ρ152-22-200-12-1002i-2i-2i2i00001-11-21i-i-ii00    complex lifted from C4xS3
ρ162-22-2002-1-12i-2i00002i-2i00-22-2110000-ii    complex lifted from C4xS3
ρ1733-1-111300-3-311-1-111-1-13-1-1001-11-100    orthogonal lifted from C2xS4
ρ1833-1-1-1-1300-3-3-1-11111113-1-100-11-1100    orthogonal lifted from C2xS4
ρ1933-1-1-1-13003311-1-1-1-1113-1-1001-11-100    orthogonal lifted from S4
ρ2033-1-11130033-1-111-1-1-1-13-1-100-11-1100    orthogonal lifted from S4
ρ213-3-11-113003i-3ii-ii-i-ii1-1-3-1100-i-iii00    complex lifted from C4xS4
ρ223-3-111-13003i-3i-ii-ii-ii-11-3-1100ii-i-i00    complex lifted from C4xS4
ρ233-3-111-1300-3i3ii-ii-ii-i-11-3-1100-i-iii00    complex lifted from C4xS4
ρ243-3-11-11300-3i3i-ii-iii-i1-1-3-1100ii-i-i00    complex lifted from C4xS4
ρ254-44-400-2-2100000000002-222-1000000    orthogonal lifted from C6.D6
ρ26444400-2-210000000000-2-2-2-21000000    orthogonal lifted from S32
ρ2766-2-200-3000022-2-20000-31100-11-1100    orthogonal lifted from S3xS4
ρ2866-2-200-30000-2-2220000-311001-11-100    orthogonal lifted from S3xS4
ρ296-6-2200-300002i-2i2i-2i000031-100ii-i-i00    complex faithful
ρ306-6-2200-30000-2i2i-2i2i000031-100-i-iii00    complex faithful

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

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

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

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

Matrix representation of Dic3:2S4 in GL5(F13)

012000
11000
001200
000120
000012
,
50000
88000
00800
00080
00008
,
10000
01000
001200
000120
00001
,
10000
01000
00100
000120
000012
,
10000
01000
00001
00100
00010
,
10000
1212000
001200
000012
000120

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

Dic3:2S4 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_2S_4
% in TeX

G:=Group("Dic3:2S4");
// GroupNames label

G:=SmallGroup(288,854);
// by ID

G=gap.SmallGroup(288,854);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,28,36,234,1684,3036,782,1777,1350]);
// 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=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*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 Dic3:2S4 in TeX

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