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## G = D4×D9order 144 = 24·32

### Direct product of D4 and D9

Aliases: D4×D9, C41D18, C36⋊C22, D363C2, C12.5D6, C222D18, D182C22, C18.5C23, Dic91C22, C92(C2×D4), C3.(S3×D4), (D4×C9)⋊2C2, (C4×D9)⋊1C2, C9⋊D41C2, (C2×C18)⋊C22, (C2×C6).2D6, (C3×D4).3S3, (C22×D9)⋊2C2, C2.6(C22×D9), C6.23(C22×S3), SmallGroup(144,41)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — D4×D9
 Chief series C1 — C3 — C9 — C18 — D18 — C22×D9 — D4×D9
 Lower central C9 — C18 — D4×D9
 Upper central C1 — C2 — D4

Generators and relations for D4×D9
G = < a,b,c,d | a4=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 355 in 81 conjugacy classes, 31 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, C23, C9, Dic3, C12, D6, C2×C6, C2×D4, D9, D9, C18, C18, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, Dic9, C36, D18, D18, D18, C2×C18, S3×D4, C4×D9, D36, C9⋊D4, D4×C9, C22×D9, D4×D9
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C22×S3, D18, S3×D4, C22×D9, D4×D9

Character table of D4×D9

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A 6B 6C 9A 9B 9C 12 18A 18B 18C 18D 18E 18F 18G 18H 18I 36A 36B 36C size 1 1 2 2 9 9 18 18 2 2 18 2 4 4 2 2 2 4 2 2 2 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 1 1 1 -1 1 -1 -1 1 1 -1 1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 -1 -1 -1 linear of order 2 ρ3 1 1 1 -1 -1 -1 -1 1 1 -1 1 1 1 -1 1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 -1 -1 -1 -1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 -1 1 1 1 -1 1 1 -1 -1 1 -1 1 1 1 1 -1 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ6 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ7 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ8 1 1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 1 1 1 1 -1 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ9 2 -2 0 0 2 -2 0 0 2 0 0 -2 0 0 2 2 2 0 -2 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 0 0 0 0 2 2 0 2 -2 -2 -1 -1 -1 2 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 orthogonal lifted from D6 ρ11 2 2 -2 2 0 0 0 0 2 -2 0 2 -2 2 -1 -1 -1 -2 -1 -1 -1 1 -1 -1 -1 1 1 1 1 1 orthogonal lifted from D6 ρ12 2 2 2 -2 0 0 0 0 2 -2 0 2 2 -2 -1 -1 -1 -2 -1 -1 -1 -1 1 1 1 -1 -1 1 1 1 orthogonal lifted from D6 ρ13 2 -2 0 0 -2 2 0 0 2 0 0 -2 0 0 2 2 2 0 -2 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 0 0 0 0 2 2 0 2 2 2 -1 -1 -1 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ15 2 2 2 -2 0 0 0 0 -1 -2 0 -1 -1 1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 ζ98+ζ9 ζ97+ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 orthogonal lifted from D18 ρ16 2 2 2 -2 0 0 0 0 -1 -2 0 -1 -1 1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 ζ97+ζ92 ζ95+ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 orthogonal lifted from D18 ρ17 2 2 2 -2 0 0 0 0 -1 -2 0 -1 -1 1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 ζ95+ζ94 ζ98+ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 orthogonal lifted from D18 ρ18 2 2 -2 2 0 0 0 0 -1 -2 0 -1 1 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -ζ98-ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 orthogonal lifted from D18 ρ19 2 2 2 2 0 0 0 0 -1 2 0 -1 -1 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ20 2 2 2 2 0 0 0 0 -1 2 0 -1 -1 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ21 2 2 -2 2 0 0 0 0 -1 -2 0 -1 1 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -ζ97-ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 orthogonal lifted from D18 ρ22 2 2 -2 -2 0 0 0 0 -1 2 0 -1 1 1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ95-ζ94 -ζ98-ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D18 ρ23 2 2 -2 2 0 0 0 0 -1 -2 0 -1 1 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -ζ95-ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 orthogonal lifted from D18 ρ24 2 2 -2 -2 0 0 0 0 -1 2 0 -1 1 1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ97-ζ92 -ζ95-ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D18 ρ25 2 2 2 2 0 0 0 0 -1 2 0 -1 -1 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ26 2 2 -2 -2 0 0 0 0 -1 2 0 -1 1 1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ98-ζ9 -ζ97-ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D18 ρ27 4 -4 0 0 0 0 0 0 4 0 0 -4 0 0 -2 -2 -2 0 2 2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ28 4 -4 0 0 0 0 0 0 -2 0 0 2 0 0 2ζ95+2ζ94 2ζ97+2ζ92 2ζ98+2ζ9 0 -2ζ97-2ζ92 -2ζ98-2ζ9 -2ζ95-2ζ94 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ29 4 -4 0 0 0 0 0 0 -2 0 0 2 0 0 2ζ98+2ζ9 2ζ95+2ζ94 2ζ97+2ζ92 0 -2ζ95-2ζ94 -2ζ97-2ζ92 -2ζ98-2ζ9 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ30 4 -4 0 0 0 0 0 0 -2 0 0 2 0 0 2ζ97+2ζ92 2ζ98+2ζ9 2ζ95+2ζ94 0 -2ζ98-2ζ9 -2ζ95-2ζ94 -2ζ97-2ζ92 0 0 0 0 0 0 0 0 0 orthogonal faithful

Smallest permutation representation of D4×D9
On 36 points
Generators in S36
(1 32 14 23)(2 33 15 24)(3 34 16 25)(4 35 17 26)(5 36 18 27)(6 28 10 19)(7 29 11 20)(8 30 12 21)(9 31 13 22)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 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 13)(2 12)(3 11)(4 10)(5 18)(6 17)(7 16)(8 15)(9 14)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(27 36)

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

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

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

D4×D9 is a maximal subgroup of
D8⋊D9  D72⋊C2  D46D18  D48D18  C36⋊D6  D18⋊D6
D4×D9 is a maximal quotient of
C222Dic18  Dic94D4  C223D36  C23.9D18  D18⋊D4  Dic9.D4  C36⋊Q8  D36⋊C4  D18.D4  C4⋊D36  D18⋊Q8  D8⋊D9  D83D9  D72⋊C2  SD16⋊D9  SD163D9  Q16⋊D9  D725C2  C232D18  C362D4  Dic9⋊D4  C36⋊D4  C36⋊D6  D18⋊D6

Matrix representation of D4×D9 in GL4(𝔽37) generated by

 0 1 0 0 36 0 0 0 0 0 36 0 0 0 0 36
,
 0 1 0 0 1 0 0 0 0 0 36 0 0 0 0 36
,
 1 0 0 0 0 1 0 0 0 0 31 20 0 0 17 11
,
 36 0 0 0 0 36 0 0 0 0 17 6 0 0 26 20
G:=sub<GL(4,GF(37))| [0,36,0,0,1,0,0,0,0,0,36,0,0,0,0,36],[0,1,0,0,1,0,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,31,17,0,0,20,11],[36,0,0,0,0,36,0,0,0,0,17,26,0,0,6,20] >;

D4×D9 in GAP, Magma, Sage, TeX

D_4\times D_9
% in TeX

G:=Group("D4xD9");
// GroupNames label

G:=SmallGroup(144,41);
// by ID

G=gap.SmallGroup(144,41);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,116,2404,208,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^9=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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