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G = SD16×D9order 288 = 25·32

Direct product of SD16 and D9

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

Aliases: SD16×D9, C85D18, Q82D18, C725C22, D4.2D18, C24.36D6, D18.13D4, C36.4C23, Dic9.4D4, D36.2C22, Dic182C22, (D4×D9).C2, (C8×D9)⋊4C2, C9⋊C86C22, (Q8×D9)⋊1C2, C92(C2×SD16), C3.(S3×SD16), C72⋊C25C2, D4.D93C2, (C3×D4).4D6, C6.92(S3×D4), C2.18(D4×D9), Q82D91C2, (C9×SD16)⋊3C2, C18.30(C2×D4), (C3×Q8).24D6, (Q8×C9)⋊1C22, C4.4(C22×D9), (C3×SD16).3S3, (C4×D9).9C22, (D4×C9).2C22, C12.43(C22×S3), SmallGroup(288,123)

Series: Derived Chief Lower central Upper central

C1C36 — SD16×D9
C1C3C9C18C36C4×D9D4×D9 — SD16×D9
C9C18C36 — SD16×D9
C1C2C4SD16

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 [×4], C3, C4, C4 [×3], C22 [×5], S3 [×3], C6, C6, C8, C8, C2×C4 [×2], D4, D4 [×2], Q8, Q8 [×2], C23, C9, Dic3 [×2], C12, C12, D6 [×4], C2×C6, C2×C8, SD16, SD16 [×3], C2×D4, C2×Q8, D9 [×2], D9, C18, C18, C3⋊C8, C24, Dic6 [×2], C4×S3 [×2], D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C2×SD16, Dic9, Dic9, C36, C36, D18, D18 [×3], 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 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], SD16 [×2], C2×D4, D9, C22×S3, C2×SD16, D18 [×3], 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 2A2B2C2D2E 3 4A4B4C4D6A6B8A8B8C8D9A9B9C12A12B18A18B18C18D18E18F24A24B36A36B36C36D36E36F72A···72F
order122222344446688889991212181818181818242436363636363672···72
size114993622418362822181822248222888444448884···4

42 irreducible representations

dim11111111222222222224444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D6SD16D9D18D18D18S3×D4S3×SD16D4×D9SD16×D9
kernelSD16×D9C8×D9C72⋊C2D4.D9Q82D9C9×SD16D4×D9Q8×D9C3×SD16Dic9D18C24C3×D4C3×Q8D9SD16C8D4Q8C6C3C2C1
# reps11111111111111433331236

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

72000
07200
006112
00670
,
1000
0100
0010
00172
,
287000
33100
0010
0001
,
0100
1000
0010
0001
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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