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G = D9×S4order 432 = 24·33

Direct product of D9 and S4

direct product, non-abelian, soluble, monomial

Aliases: D9×S4, A41D18, C9⋊S4⋊C2, (C9×S4)⋊C2, (A4×D9)⋊C2, (C2×C18)⋊D6, C91(C2×S4), (C3×S4).S3, (C9×A4)⋊C22, C3.1(S3×S4), (C22×D9)⋊S3, (C3×A4).2D6, C221(S3×D9), (C2×C6).1S32, SmallGroup(432,521)

Series: Derived Chief Lower central Upper central

C1C22C9×A4 — D9×S4
C1C22C2×C6C2×C18C9×A4A4×D9 — D9×S4
C9×A4 — D9×S4
C1

Generators and relations for D9×S4
 G = < a,b,c,d,e,f | a9=b2=c2=d2=e3=f2=1, bab=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: 1114 in 107 conjugacy classes, 17 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C9, C9, C32, Dic3, C12, A4, A4, D6, C2×C6, C2×C6, C2×D4, D9, D9, C18, C3×S3, C3⋊S3, C4×S3, D12, C3⋊D4, C3×D4, S4, S4, C2×A4, C22×S3, C3×C9, Dic9, C36, C3.A4, D18, C2×C18, C2×C18, S32, C3×A4, S3×D4, C2×S4, C3×D9, S3×C9, C9⋊S3, C4×D9, D36, C9⋊D4, D4×C9, C3.S4, C22×D9, C22×D9, C3×S4, C3⋊S4, S3×A4, S3×D9, C9×A4, D4×D9, S3×S4, C9×S4, C9⋊S4, A4×D9, D9×S4
Quotients: C1, C2, C22, S3, D6, D9, S4, D18, S32, C2×S4, S3×D9, S3×S4, D9×S4

Character table of D9×S4

 class 12A2B2C2D2E3A3B3C4A4B6A6B6C9A9B9C9D9E9F1218A18B18C18D18E18F36A36B36C
 size 1369275428166546127222216161612666121212121212
ρ1111111111111111111111111111111    trivial
ρ211-1-1-11111-111-1-1111111-1111-1-1-1-1-1-1    linear of order 2
ρ311-111-1111-1-11-11111111-1111-1-1-1-1-1-1    linear of order 2
ρ4111-1-1-11111-111-11111111111111111    linear of order 2
ρ5220-2-202-1-100201222-1-1-10222000000    orthogonal lifted from D6
ρ62202202-1-10020-1222-1-1-10222000000    orthogonal lifted from S3
ρ722200022220220-1-1-1-1-1-12-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ822-2000222-202-20-1-1-1-1-1-1-2-1-1-1111111    orthogonal lifted from D6
ρ922-2000-12-1-20-110ζ9594ζ989ζ9792ζ9594ζ9792ζ9891ζ9792ζ989ζ95949899792959497929594989    orthogonal lifted from D18
ρ10222000-12-120-1-10ζ9594ζ989ζ9792ζ9594ζ9792ζ989-1ζ9792ζ989ζ9594ζ989ζ9792ζ9594ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ1122-2000-12-1-20-110ζ989ζ9792ζ9594ζ989ζ9594ζ97921ζ9594ζ9792ζ9899792959498995949899792    orthogonal lifted from D18
ρ12222000-12-120-1-10ζ9792ζ9594ζ989ζ9792ζ989ζ9594-1ζ989ζ9594ζ9792ζ9594ζ989ζ9792ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ1322-2000-12-1-20-110ζ9792ζ9594ζ989ζ9792ζ989ζ95941ζ989ζ9594ζ97929594989979298997929594    orthogonal lifted from D18
ρ14222000-12-120-1-10ζ989ζ9792ζ9594ζ989ζ9594ζ9792-1ζ9594ζ9792ζ989ζ9792ζ9594ζ989ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ153-1-1-3113001-1-1-103330001-1-1-1-1-1-1111    orthogonal lifted from C2×S4
ρ163-1-13-1-130011-1-103330001-1-1-1-1-1-1111    orthogonal lifted from S4
ρ173-11-31-1300-11-110333000-1-1-1-1111-1-1-1    orthogonal lifted from C2×S4
ρ183-113-11300-1-1-110333000-1-1-1-1111-1-1-1    orthogonal lifted from S4
ρ194400004-2-200400-2-2-21110-2-2-2000000    orthogonal lifted from S32
ρ20440000-2-2100-20095+2ζ9498+2ζ997+2ζ9295949792989097+2ζ9298+2ζ995+2ζ94000000    orthogonal lifted from S3×D9
ρ21440000-2-2100-20098+2ζ997+2ζ9295+2ζ9498995949792095+2ζ9497+2ζ9298+2ζ9000000    orthogonal lifted from S3×D9
ρ22440000-2-2100-20097+2ζ9295+2ζ9498+2ζ997929899594098+2ζ995+2ζ9497+2ζ92000000    orthogonal lifted from S3×D9
ρ236-2-200060020-2-20-3-3-30002111111-1-1-1    orthogonal lifted from S3×S4
ρ246-22000600-20-220-3-3-3000-2111-1-1-1111    orthogonal lifted from S3×S4
ρ256-2-2000-3002011097+3ζ9295+3ζ9498+3ζ9000-19899594979295949899792ζ989ζ9792ζ9594    orthogonal faithful
ρ266-22000-300-201-1097+3ζ9295+3ζ9498+3ζ9000198995949792ζ9594ζ989ζ979298997929594    orthogonal faithful
ρ276-2-2000-3002011098+3ζ997+3ζ9295+3ζ94000-19594979298997929594989ζ9594ζ989ζ9792    orthogonal faithful
ρ286-22000-300-201-1098+3ζ997+3ζ9295+3ζ94000195949792989ζ9792ζ9594ζ98995949899792    orthogonal faithful
ρ296-22000-300-201-1095+3ζ9498+3ζ997+3ζ92000197929899594ζ989ζ9792ζ959497929594989    orthogonal faithful
ρ306-2-2000-3002011095+3ζ9498+3ζ997+3ζ92000-19792989959498997929594ζ9792ζ9594ζ989    orthogonal faithful

Smallest permutation representation of D9×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 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 25)(20 24)(21 23)(26 27)(28 36)(29 35)(30 34)(31 33)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 23)(11 24)(12 25)(13 26)(14 27)(15 19)(16 20)(17 21)(18 22)
(1 27)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 33)(11 34)(12 35)(13 36)(14 28)(15 29)(16 30)(17 31)(18 32)
(10 33 23)(11 34 24)(12 35 25)(13 36 26)(14 28 27)(15 29 19)(16 30 20)(17 31 21)(18 32 22)
(10 33)(11 34)(12 35)(13 36)(14 28)(15 29)(16 30)(17 31)(18 32)

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

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

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

Matrix representation of D9×S4 in GL5(𝔽37)

1711000
266000
00100
00010
00001
,
2026000
617000
00100
00010
00001
,
10000
01000
00001
00363636
00100
,
10000
01000
00363636
00001
00010
,
10000
01000
00100
00363636
00010
,
360000
036000
00100
00001
00010

G:=sub<GL(5,GF(37))| [17,26,0,0,0,11,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[20,6,0,0,0,26,17,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,36,1,0,0,0,36,0,0,0,1,36,0],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,36,0,1,0,0,36,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,36,0,0,0,0,36,1,0,0,0,36,0],[36,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

D9×S4 in GAP, Magma, Sage, TeX

D_9\times S_4
% in TeX

G:=Group("D9xS4");
// GroupNames label

G:=SmallGroup(432,521);
// by ID

G=gap.SmallGroup(432,521);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,93,1683,192,2524,4548,2287,2659,3989]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^9=b^2=c^2=d^2=e^3=f^2=1,b*a*b=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 D9×S4 in TeX

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