direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: SD16×D9, C8⋊5D18, Q8⋊2D18, C72⋊5C22, D4.2D18, C24.36D6, D18.13D4, C36.4C23, Dic9.4D4, D36.2C22, Dic18⋊2C22, (D4×D9).C2, (C8×D9)⋊4C2, C9⋊C8⋊6C22, (Q8×D9)⋊1C2, C9⋊2(C2×SD16), C3.(S3×SD16), C72⋊C2⋊5C2, D4.D9⋊3C2, (C3×D4).4D6, C6.92(S3×D4), C2.18(D4×D9), Q8⋊2D9⋊1C2, (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
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, Q8⋊2S3, 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, Q8⋊2D9, 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
►On 72 points
(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