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G = D8×D9  order 288 = 25·32

Direct product of D8 and D9

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

Aliases: D8×D9, C8⋊4D18, D72⋊4C2, D4⋊1D18, C72⋊2C22, C24.10D6, D18.12D4, D36⋊1C22, C36.1C23, Dic9.3D4, C9⋊2(C2×D8), C3.(S3×D8), D4⋊D9⋊1C2, (C8×D9)⋊1C2, (D4×D9)⋊1C2, (C9×D8)⋊2C2, C9⋊C8⋊5C22, (C3×D4).1D6, (C3×D8).3S3, C6.89(S3×D4), C2.15(D4×D9), C18.27(C2×D4), (D4×C9)⋊1C22, C4.1(C22×D9), (C4×D9).7C22, C12.40(C22×S3), SmallGroup(288,120)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C36 — D8×D9
Chief series
C1 — C3 — C9 — C18 — C36 — C4×D9 — D4×D9 — D8×D9
Lower central
C9 — C18 — C36 — D8×D9
Upper central
C1 — C2 — C4 — D8

Generators and relations for D8×D9
 G = < a,b,c,d | a8=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 720 in 114 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, C23, C9, Dic3, C12, D6, C2×C6, C2×C8, D8, D8, C2×D4, D9, D9, C18, C18, C3⋊C8, C24, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C2×D8, Dic9, C36, D18, D18, C2×C18, S3×C8, D24, D4⋊S3, C3×D8, S3×D4, C9⋊C8, C72, C4×D9, D36, C9⋊D4, D4×C9, C22×D9, S3×D8, C8×D9, D72, D4⋊D9, C9×D8, D4×D9, D8×D9
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, D9, C22×S3, C2×D8, D18, S3×D4, C22×D9, S3×D8, D4×D9, D8×D9

Smallest permutation representation of D8×D9
►On 72 points
Generators in S72
(1 41 23 68 14 50 32 59)(2 42 24 69 15 51 33 60)(3 43 25 70 16 52 34 61)(4 44 26 71 17 53 35 62)(5 45 27 72 18 54 36 63)(6 37 19 64 10 46 28 55)(7 38 20 65 11 47 29 56)(8 39 21 66 12 48 30 57)(9 40 22 67 13 49 31 58)

(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)

(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)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)

(1 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)(28 35)(29 34)(30 33)(31 32)(37 44)(38 43)(39 42)(40 41)(46 53)(47 52)(48 51)(49 50)(55 62)(56 61)(57 60)(58 59)(64 71)(65 70)(66 69)(67 68)
 
G:=sub<Sym(72)| (1,41,23,68,14,50,32,59)(2,42,24,69,15,51,33,60)(3,43,25,70,16,52,34,61)(4,44,26,71,17,53,35,62)(5,45,27,72,18,54,36,63)(6,37,19,64,10,46,28,55)(7,38,20,65,11,47,29,56)(8,39,21,66,12,48,30,57)(9,40,22,67,13,49,31,58), (19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(37,44)(38,43)(39,42)(40,41)(46,53)(47,52)(48,51)(49,50)(55,62)(56,61)(57,60)(58,59)(64,71)(65,70)(66,69)(67,68)>;
 
G:=Group( (1,41,23,68,14,50,32,59)(2,42,24,69,15,51,33,60)(3,43,25,70,16,52,34,61)(4,44,26,71,17,53,35,62)(5,45,27,72,18,54,36,63)(6,37,19,64,10,46,28,55)(7,38,20,65,11,47,29,56)(8,39,21,66,12,48,30,57)(9,40,22,67,13,49,31,58), (19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(37,44)(38,43)(39,42)(40,41)(46,53)(47,52)(48,51)(49,50)(55,62)(56,61)(57,60)(58,59)(64,71)(65,70)(66,69)(67,68) );
 
G=PermutationGroup([[(1,41,23,68,14,50,32,59),(2,42,24,69,15,51,33,60),(3,43,25,70,16,52,34,61),(4,44,26,71,17,53,35,62),(5,45,27,72,18,54,36,63),(6,37,19,64,10,46,28,55),(7,38,20,65,11,47,29,56),(8,39,21,66,12,48,30,57),(9,40,22,67,13,49,31,58)], [(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72)], [(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),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,26),(20,25),(21,24),(22,23),(28,35),(29,34),(30,33),(31,32),(37,44),(38,43),(39,42),(40,41),(46,53),(47,52),(48,51),(49,50),(55,62),(56,61),(57,60),(58,59),(64,71),(65,70),(66,69),(67,68)]])
 

42 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B6A6B6C8A8B8C8D9A9B9C 12 18A18B18C18D···18I24A24B36A36B36C72A···72F
order1222222234466688889991218181818···18242436363672···72
size1144993636221828822181822242228···8444444···4

42 irreducible representations

dim1111112222222224444
type+++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D8D9D18D18S3×D4S3×D8D4×D9D8×D9
kernelD8×D9C8×D9D72D4⋊D9C9×D8D4×D9C3×D8Dic9D18C24C3×D4D9D8C8D4C6C3C2C1
# reps1112121111243361236

Matrix representation of D8×D9 ►in GL4(𝔽73) generated by

72000
07200
003225
00350
,
1000
0100
0010
002572
,
453100
42300
0010
0001
,
42300
453100
0010
0001
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,32,35,0,0,25,0],[1,0,0,0,0,1,0,0,0,0,1,25,0,0,0,72],[45,42,0,0,31,3,0,0,0,0,1,0,0,0,0,1],[42,45,0,0,3,31,0,0,0,0,1,0,0,0,0,1] >;
 

D8×D9 in GAP, Magma, Sage, TeX 

D_8\times D_9
 
% in TeX
 
G:=Group("D8xD9");
 
// GroupNames label
 
G:=SmallGroup(288,120);
 
// by ID
 
G=gap.SmallGroup(288,120);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,135,346,185,80,6725,292,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=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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