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## G = SD16×D9order 288 = 25·32

### Direct product of SD16 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — SD16×D9
 Chief series C1 — C3 — C9 — C18 — C36 — C4×D9 — D4×D9 — SD16×D9
 Lower central C9 — C18 — C36 — SD16×D9
 Upper central C1 — C2 — C4 — SD16

Generators and relations for SD16×D9
G = < a,b,c,d | a8=b2=c9=d2=1, bab=a3, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 552 in 102 conjugacy classes, 36 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, C9, Dic3, C12, C12, D6, C2×C6, C2×C8, SD16, SD16, C2×D4, C2×Q8, D9, D9, C18, C18, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C2×SD16, Dic9, Dic9, C36, C36, D18, D18, C2×C18, S3×C8, C24⋊C2, D4.S3, Q82S3, C3×SD16, S3×D4, S3×Q8, C9⋊C8, C72, Dic18, Dic18, C4×D9, C4×D9, D36, C9⋊D4, D4×C9, Q8×C9, C22×D9, S3×SD16, C8×D9, C72⋊C2, D4.D9, Q82D9, C9×SD16, D4×D9, Q8×D9, SD16×D9
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, D9, C22×S3, C2×SD16, D18, S3×D4, C22×D9, S3×SD16, D4×D9, SD16×D9

Smallest permutation representation of SD16×D9
On 72 points
Generators in S72
(1 68 23 50 14 59 32 41)(2 69 24 51 15 60 33 42)(3 70 25 52 16 61 34 43)(4 71 26 53 17 62 35 44)(5 72 27 54 18 63 36 45)(6 64 19 46 10 55 28 37)(7 65 20 47 11 56 29 38)(8 66 21 48 12 57 30 39)(9 67 22 49 13 58 31 40)
(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)| (1,68,23,50,14,59,32,41)(2,69,24,51,15,60,33,42)(3,70,25,52,16,61,34,43)(4,71,26,53,17,62,35,44)(5,72,27,54,18,63,36,45)(6,64,19,46,10,55,28,37)(7,65,20,47,11,56,29,38)(8,66,21,48,12,57,30,39)(9,67,22,49,13,58,31,40), (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( (1,68,23,50,14,59,32,41)(2,69,24,51,15,60,33,42)(3,70,25,52,16,61,34,43)(4,71,26,53,17,62,35,44)(5,72,27,54,18,63,36,45)(6,64,19,46,10,55,28,37)(7,65,20,47,11,56,29,38)(8,66,21,48,12,57,30,39)(9,67,22,49,13,58,31,40), (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([[(1,68,23,50,14,59,32,41),(2,69,24,51,15,60,33,42),(3,70,25,52,16,61,34,43),(4,71,26,53,17,62,35,44),(5,72,27,54,18,63,36,45),(6,64,19,46,10,55,28,37),(7,65,20,47,11,56,29,38),(8,66,21,48,12,57,30,39),(9,67,22,49,13,58,31,40)], [(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 3 4A 4B 4C 4D 6A 6B 8A 8B 8C 8D 9A 9B 9C 12A 12B 18A 18B 18C 18D 18E 18F 24A 24B 36A 36B 36C 36D 36E 36F 72A ··· 72F order 1 2 2 2 2 2 3 4 4 4 4 6 6 8 8 8 8 9 9 9 12 12 18 18 18 18 18 18 24 24 36 36 36 36 36 36 72 ··· 72 size 1 1 4 9 9 36 2 2 4 18 36 2 8 2 2 18 18 2 2 2 4 8 2 2 2 8 8 8 4 4 4 4 4 8 8 8 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 SD16 D9 D18 D18 D18 S3×D4 S3×SD16 D4×D9 SD16×D9 kernel SD16×D9 C8×D9 C72⋊C2 D4.D9 Q8⋊2D9 C9×SD16 D4×D9 Q8×D9 C3×SD16 Dic9 D18 C24 C3×D4 C3×Q8 D9 SD16 C8 D4 Q8 C6 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 3 3 3 3 1 2 3 6

Matrix representation of SD16×D9 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 61 12 0 0 67 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 72
,
 28 70 0 0 3 31 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,61,67,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,72],[28,3,0,0,70,31,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

SD16×D9 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times D_9
% in TeX

G:=Group("SD16xD9");
// GroupNames label

G:=SmallGroup(288,123);
// by ID

G=gap.SmallGroup(288,123);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,135,100,346,185,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^9=d^2=1,b*a*b=a^3,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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