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G = Dic32S4order 288 = 25·32

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

non-abelian, soluble, monomial

Aliases: Dic32S4, C3⋊S4⋊C4, C31(C4×S4), A4⋊C42S3, A41(C4×S3), C2.2(S3×S4), C23.3S32, C6.11(C2×S4), (C2×A4).3D6, (Dic3×A4)⋊3C2, (C22×C6).3D6, (C6×A4).3C22, C22⋊(C6.D6), (C22×Dic3)⋊2S3, (C2×C3⋊S4).C2, (C2×C6)⋊2(C4×S3), (C3×A4⋊C4)⋊2C2, (C3×A4)⋊2(C2×C4), SmallGroup(288,854)

Series: Derived Chief Lower central Upper central

C1C22C3×A4 — Dic32S4
C1C22C2×C6C3×A4C6×A4Dic3×A4 — Dic32S4
C3×A4 — Dic32S4
C1C2

Generators and relations for Dic32S4
 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 [×4], C3, C3 [×2], C4 [×6], C22, C22 [×6], S3 [×6], C6, C6 [×4], C2×C4 [×7], D4 [×4], C23, C23, C32, Dic3, Dic3 [×4], C12 [×3], A4, A4, D6 [×6], C2×C6, C2×C6 [×2], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3⋊S3 [×2], C3×C6, C4×S3 [×3], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×2], S4 [×4], C2×A4, C2×A4, C22×S3, C22×C6, C4×D4, C3×Dic3 [×2], C3×A4, C2×C3⋊S3, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, A4⋊C4, C4×A4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C2×S4 [×2], C6.D6, C3⋊S4 [×2], C6×A4, Dic34D4, C4×S4, C3×A4⋊C4, Dic3×A4, C2×C3⋊S4, Dic32S4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D6 [×2], C4×S3 [×2], S4, S32, C2×S4, C6.D6, C4×S4, S3×S4, Dic32S4

Character table of Dic32S4

 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 C4×S3
ρ142-22-200-12-100-2i2i2i-2i00001-11-21-iii-i00    complex lifted from C4×S3
ρ152-22-200-12-1002i-2i-2i2i00001-11-21i-i-ii00    complex lifted from C4×S3
ρ162-22-2002-1-12i-2i00002i-2i00-22-2110000-ii    complex lifted from C4×S3
ρ1733-1-111300-3-311-1-111-1-13-1-1001-11-100    orthogonal lifted from C2×S4
ρ1833-1-1-1-1300-3-3-1-11111113-1-100-11-1100    orthogonal lifted from C2×S4
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 C4×S4
ρ223-3-111-13003i-3i-ii-ii-ii-11-3-1100ii-i-i00    complex lifted from C4×S4
ρ233-3-111-1300-3i3ii-ii-ii-i-11-3-1100-i-iii00    complex lifted from C4×S4
ρ243-3-11-11300-3i3i-ii-iii-i1-1-3-1100ii-i-i00    complex lifted from C4×S4
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 S3×S4
ρ2866-2-200-30000-2-2220000-311001-11-100    orthogonal lifted from S3×S4
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 Dic32S4
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 Dic32S4 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] >;

Dic32S4 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 Dic32S4 in TeX

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