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G = D8⋊D9order 288 = 25·32

2nd semidirect product of D8 and D9 acting via D9/C9=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D82D9, C82D18, D42D18, C724C22, D18.6D4, C24.33D6, C36.2C23, Dic9.8D4, D36.1C22, Dic181C22, D4⋊D92C2, (D4×D9)⋊2C2, (C9×D8)⋊4C2, C9⋊C81C22, C72⋊C23C2, C8⋊D93C2, C92(C8⋊C22), D4.D91C2, C3.(D8⋊S3), (C3×D8).6S3, (C3×D4).2D6, C6.90(S3×D4), C2.16(D4×D9), D42D91C2, C18.28(C2×D4), (D4×C9)⋊2C22, C4.2(C22×D9), (C4×D9).1C22, C12.41(C22×S3), SmallGroup(288,121)

Series: Derived Chief Lower central Upper central

C1C36 — D8⋊D9
C1C3C9C18C36C4×D9D4×D9 — D8⋊D9
C9C18C36 — D8⋊D9
C1C2C4D8

Generators and relations for D8⋊D9
 G = < a,b,c,d | a8=b2=c9=d2=1, bab=a-1, ac=ca, dad=a5, bc=cb, bd=db, dcd=c-1 >

Subgroups: 564 in 102 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], S3 [×2], C6, C6 [×2], C8, C8, C2×C4 [×2], D4 [×2], D4 [×3], Q8, C23, C9, Dic3 [×2], C12, D6 [×4], C2×C6 [×2], M4(2), D8, D8, SD16 [×2], C2×D4, C4○D4, D9 [×2], C18, C18 [×2], C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C8⋊C22, Dic9, Dic9, C36, D18, D18 [×3], C2×C18 [×2], C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D42S3, C9⋊C8, C72, Dic18, C4×D9, D36, C2×Dic9, C9⋊D4 [×2], D4×C9 [×2], C22×D9, D8⋊S3, C8⋊D9, C72⋊C2, D4.D9, D4⋊D9, C9×D8, D4×D9, D42D9, D8⋊D9
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, C22×S3, C8⋊C22, D18 [×3], S3×D4, C22×D9, D8⋊S3, 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 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(45 54)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(63 72)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C6A6B6C8A8B9A9B9C 12 18A18B18C18D···18I24A24B36A36B36C72A···72F
order1222223444666889991218181818···18242436363672···72
size1144183622183628843622242228···8444444···4

39 irreducible representations

dim111111112222222244444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D9D18D18C8⋊C22S3×D4D8⋊S3D4×D9D8⋊D9
kernelD8⋊D9C8⋊D9C72⋊C2D4.D9D4⋊D9C9×D8D4×D9D42D9C3×D8Dic9D18C24C3×D4D8C8D4C9C6C3C2C1
# reps111111111111233611236

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

31624211
11426231
31623162
11421142
,
1000
0100
00720
00072
,
33100
424500
00331
004245
,
33100
287000
00331
002870
G:=sub<GL(4,GF(73))| [31,11,31,11,62,42,62,42,42,62,31,11,11,31,62,42],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[3,42,0,0,31,45,0,0,0,0,3,42,0,0,31,45],[3,28,0,0,31,70,0,0,0,0,3,28,0,0,31,70] >;

D8⋊D9 in GAP, Magma, Sage, TeX

D_8\rtimes D_9
% in TeX

G:=Group("D8:D9");
// GroupNames label

G:=SmallGroup(288,121);
// by ID

G=gap.SmallGroup(288,121);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,346,185,80,6725,292,9414]);
// 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,d*a*d=a^5,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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