Copied to
clipboard

G = SD16xD9order 288 = 25·32

Direct product of SD16 and D9

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

Aliases: SD16xD9, C8:5D18, Q8:2D18, C72:5C22, D4.2D18, C24.36D6, D18.13D4, C36.4C23, Dic9.4D4, D36.2C22, Dic18:2C22, (D4xD9).C2, (C8xD9):4C2, C9:C8:6C22, (Q8xD9):1C2, C9:2(C2xSD16), C3.(S3xSD16), C72:C2:5C2, D4.D9:3C2, (C3xD4).4D6, C6.92(S3xD4), C2.18(D4xD9), Q8:2D9:1C2, (C9xSD16):3C2, C18.30(C2xD4), (C3xQ8).24D6, (Q8xC9):1C22, C4.4(C22xD9), (C3xSD16).3S3, (C4xD9).9C22, (D4xC9).2C22, C12.43(C22xS3), SmallGroup(288,123)

Series: Derived Chief Lower central Upper central

C1C36 — SD16xD9
C1C3C9C18C36C4xD9D4xD9 — SD16xD9
C9C18C36 — SD16xD9
C1C2C4SD16

Generators and relations for SD16xD9
 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, C2xC4, D4, D4, Q8, Q8, C23, C9, Dic3, C12, C12, D6, C2xC6, C2xC8, SD16, SD16, C2xD4, C2xQ8, D9, D9, C18, C18, C3:C8, C24, Dic6, C4xS3, D12, C3:D4, C3xD4, C3xQ8, C22xS3, C2xSD16, Dic9, Dic9, C36, C36, D18, D18, C2xC18, S3xC8, C24:C2, D4.S3, Q8:2S3, C3xSD16, S3xD4, S3xQ8, C9:C8, C72, Dic18, Dic18, C4xD9, C4xD9, D36, C9:D4, D4xC9, Q8xC9, C22xD9, S3xSD16, C8xD9, C72:C2, D4.D9, Q8:2D9, C9xSD16, D4xD9, Q8xD9, SD16xD9
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2xD4, D9, C22xS3, C2xSD16, D18, S3xD4, C22xD9, S3xSD16, D4xD9, SD16xD9

Smallest permutation representation of SD16xD9
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++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D6SD16D9D18D18D18S3xD4S3xSD16D4xD9SD16xD9
kernelSD16xD9C8xD9C72:C2D4.D9Q8:2D9C9xSD16D4xD9Q8xD9C3xSD16Dic9D18C24C3xD4C3xQ8D9SD16C8D4Q8C6C3C2C1
# reps11111111111111433331236

Matrix representation of SD16xD9 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] >;

SD16xD9 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

׿
x
:
Z
F
o
wr
Q
<