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## G = D8×D9order 288 = 25·32

### Direct product of D8 and D9

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 [×6], C3, C4, C4, C22 [×9], S3 [×4], C6, C6 [×2], C8, C8, C2×C4, D4 [×2], D4 [×4], C23 [×2], C9, Dic3, C12, D6 [×7], C2×C6 [×2], C2×C8, D8, D8 [×3], C2×D4 [×2], D9 [×2], D9 [×2], C18, C18 [×2], C3⋊C8, C24, C4×S3, D12 [×2], C3⋊D4 [×2], C3×D4 [×2], C22×S3 [×2], C2×D8, Dic9, C36, D18, D18 [×6], C2×C18 [×2], S3×C8, D24, D4⋊S3 [×2], C3×D8, S3×D4 [×2], C9⋊C8, C72, C4×D9, D36 [×2], C9⋊D4 [×2], D4×C9 [×2], C22×D9 [×2], S3×D8, C8×D9, D72, D4⋊D9 [×2], C9×D8, D4×D9 [×2], D8×D9
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], D8 [×2], C2×D4, D9, C22×S3, C2×D8, D18 [×3], 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 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A 6B 6C 8A 8B 8C 8D 9A 9B 9C 12 18A 18B 18C 18D ··· 18I 24A 24B 36A 36B 36C 72A ··· 72F order 1 2 2 2 2 2 2 2 3 4 4 6 6 6 8 8 8 8 9 9 9 12 18 18 18 18 ··· 18 24 24 36 36 36 72 ··· 72 size 1 1 4 4 9 9 36 36 2 2 18 2 8 8 2 2 18 18 2 2 2 4 2 2 2 8 ··· 8 4 4 4 4 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D8 D9 D18 D18 S3×D4 S3×D8 D4×D9 D8×D9 kernel D8×D9 C8×D9 D72 D4⋊D9 C9×D8 D4×D9 C3×D8 Dic9 D18 C24 C3×D4 D9 D8 C8 D4 C6 C3 C2 C1 # reps 1 1 1 2 1 2 1 1 1 1 2 4 3 3 6 1 2 3 6

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

 72 0 0 0 0 72 0 0 0 0 32 25 0 0 35 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 25 72
,
 45 31 0 0 42 3 0 0 0 0 1 0 0 0 0 1
,
 42 3 0 0 45 31 0 0 0 0 1 0 0 0 0 1
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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