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G = D4×D9order 144 = 24·32

Direct product of D4 and D9

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

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

C1C18 — D4×D9
C1C3C9C18D18C22×D9 — D4×D9
C9C18 — D4×D9
C1C2D4

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 [×6], C3, C4, C4, C22 [×2], C22 [×7], S3 [×4], C6, C6 [×2], C2×C4, D4, D4 [×3], C23 [×2], C9, Dic3, C12, D6 [×7], C2×C6 [×2], C2×D4, D9 [×2], D9 [×2], C18, C18 [×2], C4×S3, D12, C3⋊D4 [×2], C3×D4, C22×S3 [×2], Dic9, C36, D18, D18 [×2], D18 [×4], C2×C18 [×2], S3×D4, C4×D9, D36, C9⋊D4 [×2], D4×C9, C22×D9 [×2], D4×D9
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, C22×S3, D18 [×3], S3×D4, C22×D9, D4×D9

Character table of D4×D9

 class 12A2B2C2D2E2F2G34A4B6A6B6C9A9B9C1218A18B18C18D18E18F18G18H18I36A36B36C
 size 112299181822182442224222444444444
ρ1111111111111111111111111111111    trivial
ρ2111-1111-11-1-111-1111-11111-1-1-111-1-1-1    linear of order 2
ρ3111-1-1-1-111-1111-1111-11111-1-1-111-1-1-1    linear of order 2
ρ41111-1-1-1-111-11111111111111111111    linear of order 2
ρ511-1111-111-1-11-11111-1111-1111-1-1-1-1-1    linear of order 2
ρ611-1-111-1-11111-1-11111111-1-1-1-1-1-1111    linear of order 2
ρ711-1-1-1-11111-11-1-11111111-1-1-1-1-1-1111    linear of order 2
ρ811-11-1-11-11-111-11111-1111-1111-1-1-1-1-1    linear of order 2
ρ92-2002-200200-2002220-2-2-2000000000    orthogonal lifted from D4
ρ1022-2-200002202-2-2-1-1-12-1-1-1111111-1-1-1    orthogonal lifted from D6
ρ1122-2200002-202-22-1-1-1-2-1-1-11-1-1-111111    orthogonal lifted from D6
ρ12222-200002-2022-2-1-1-1-2-1-1-1-1111-1-1111    orthogonal lifted from D6
ρ132-200-2200200-2002220-2-2-2000000000    orthogonal lifted from D4
ρ1422220000220222-1-1-12-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ15222-20000-1-20-1-11ζ9792ζ989ζ95941ζ989ζ9594ζ9792ζ959495949899792ζ989ζ979297929594989    orthogonal lifted from D18
ρ16222-20000-1-20-1-11ζ9594ζ9792ζ9891ζ9792ζ989ζ9594ζ98998997929594ζ9792ζ959495949899792    orthogonal lifted from D18
ρ17222-20000-1-20-1-11ζ989ζ9594ζ97921ζ9594ζ9792ζ989ζ979297929594989ζ9594ζ98998997929594    orthogonal lifted from D18
ρ1822-220000-1-20-11-1ζ9594ζ9792ζ9891ζ9792ζ989ζ9594989ζ989ζ9792ζ95949792959495949899792    orthogonal lifted from D18
ρ1922220000-120-1-1-1ζ989ζ9594ζ9792-1ζ9594ζ9792ζ989ζ9792ζ9792ζ9594ζ989ζ9594ζ989ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ2022220000-120-1-1-1ζ9792ζ989ζ9594-1ζ989ζ9594ζ9792ζ9594ζ9594ζ989ζ9792ζ989ζ9792ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ2122-220000-1-20-11-1ζ989ζ9594ζ97921ζ9594ζ9792ζ9899792ζ9792ζ9594ζ989959498998997929594    orthogonal lifted from D18
ρ2222-2-20000-120-111ζ989ζ9594ζ9792-1ζ9594ζ9792ζ9899792979295949899594989ζ989ζ9792ζ9594    orthogonal lifted from D18
ρ2322-220000-1-20-11-1ζ9792ζ989ζ95941ζ989ζ9594ζ97929594ζ9594ζ989ζ9792989979297929594989    orthogonal lifted from D18
ρ2422-2-20000-120-111ζ9594ζ9792ζ989-1ζ9792ζ989ζ95949899899792959497929594ζ9594ζ989ζ9792    orthogonal lifted from D18
ρ2522220000-120-1-1-1ζ9594ζ9792ζ989-1ζ9792ζ989ζ9594ζ989ζ989ζ9792ζ9594ζ9792ζ9594ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ2622-2-20000-120-111ζ9792ζ989ζ9594-1ζ989ζ9594ζ97929594959498997929899792ζ9792ζ9594ζ989    orthogonal lifted from D18
ρ274-4000000400-400-2-2-20222000000000    orthogonal lifted from S3×D4
ρ284-4000000-20020095+2ζ9497+2ζ9298+2ζ90-2ζ97-2ζ92-2ζ98-2ζ9-2ζ95-2ζ94000000000    orthogonal faithful
ρ294-4000000-20020098+2ζ995+2ζ9497+2ζ920-2ζ95-2ζ94-2ζ97-2ζ92-2ζ98-2ζ9000000000    orthogonal faithful
ρ304-4000000-20020097+2ζ9298+2ζ995+2ζ940-2ζ98-2ζ9-2ζ95-2ζ94-2ζ97-2ζ92000000000    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

0100
36000
00360
00036
,
0100
1000
00360
00036
,
1000
0100
003120
001711
,
36000
03600
00176
002620
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

Export

Character table of D4×D9 in TeX

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