Copied to
clipboard

G = Q82D9order 144 = 24·32

The semidirect product of Q8 and D9 acting via D9/C9=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q82D9, C93SD16, C4.4D18, C12.4D6, D36.2C2, C18.10D4, C36.4C22, C9⋊C83C2, (Q8×C9)⋊1C2, (C3×Q8).6S3, C2.7(C9⋊D4), C3.(Q82S3), C6.17(C3⋊D4), SmallGroup(144,18)

Series: Derived Chief Lower central Upper central

C1C36 — Q82D9
C1C3C9C18C36D36 — Q82D9
C9C18C36 — Q82D9
C1C2C4Q8

Generators and relations for Q82D9
 G = < a,b,c,d | a4=c9=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >

36C2
2C4
18C22
12S3
9D4
9C8
2C12
6D6
4D9
9SD16
3C3⋊C8
3D12
2D18
2C36
3Q82S3

Character table of Q82D9

 class 12A2B34A4B68A8B9A9B9C12A12B12C18A18B18C36A36B36C36D36E36F36G36H36I
 size 113622421818222444222444444444
ρ1111111111111111111111111111    trivial
ρ211-11111-1-1111111111111111111    linear of order 2
ρ311111-11-1-11111-1-1111-1-1-1-1-1-1111    linear of order 2
ρ411-111-11111111-1-1111-1-1-1-1-1-1111    linear of order 2
ρ5220222200-1-1-1222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ62202-20200222-200222000000-2-2-2    orthogonal lifted from D4
ρ722022-2200-1-1-12-2-2-1-1-1111111-1-1-1    orthogonal lifted from D6
ρ8220-12-2-100ζ9594ζ989ζ9792-111ζ989ζ9792ζ95949792959498995949792989ζ9792ζ9594ζ989    orthogonal lifted from D18
ρ9220-122-100ζ9594ζ989ζ9792-1-1-1ζ989ζ9792ζ9594ζ9792ζ9594ζ989ζ9594ζ9792ζ989ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ10220-12-2-100ζ989ζ9792ζ9594-111ζ9792ζ9594ζ9899594989979298995949792ζ9594ζ989ζ9792    orthogonal lifted from D18
ρ11220-12-2-100ζ9792ζ9594ζ989-111ζ9594ζ989ζ97929899792959497929899594ζ989ζ9792ζ9594    orthogonal lifted from D18
ρ12220-122-100ζ989ζ9792ζ9594-1-1-1ζ9792ζ9594ζ989ζ9594ζ989ζ9792ζ989ζ9594ζ9792ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ13220-122-100ζ9792ζ9594ζ989-1-1-1ζ9594ζ989ζ9792ζ989ζ9792ζ9594ζ9792ζ989ζ9594ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ142-20200-2-2--2222000-2-2-2000000000    complex lifted from SD16
ρ152202-20200-1-1-1-200-1-1-1--3--3--3-3-3-3111    complex lifted from C3⋊D4
ρ16220-1-20-100ζ9792ζ9594ζ9891-3--3ζ9594ζ989ζ9792ζ9899792ζ9594ζ9792989959498997929594    complex lifted from C9⋊D4
ρ17220-1-20-100ζ989ζ9792ζ95941-3--3ζ9792ζ9594ζ989ζ9594ζ98997929899594ζ979295949899792    complex lifted from C9⋊D4
ρ18220-1-20-100ζ9594ζ989ζ97921--3-3ζ989ζ9792ζ9594ζ97929594989ζ95949792ζ98997929594989    complex lifted from C9⋊D4
ρ192-20200-2--2-2222000-2-2-2000000000    complex lifted from SD16
ρ202202-20200-1-1-1-200-1-1-1-3-3-3--3--3--3111    complex lifted from C3⋊D4
ρ21220-1-20-100ζ989ζ9792ζ95941--3-3ζ9792ζ9594ζ9899594989ζ9792ζ989ζ9594979295949899792    complex lifted from C9⋊D4
ρ22220-1-20-100ζ9594ζ989ζ97921-3--3ζ989ζ9792ζ95949792ζ9594ζ9899594ζ979298997929594989    complex lifted from C9⋊D4
ρ23220-1-20-100ζ9792ζ9594ζ9891--3-3ζ9594ζ989ζ9792989ζ979295949792ζ989ζ959498997929594    complex lifted from C9⋊D4
ρ244-40400-400-2-2-2000222000000000    orthogonal lifted from Q82S3
ρ254-40-20020095+2ζ9498+2ζ997+2ζ92000-2ζ98-2ζ9-2ζ97-2ζ92-2ζ95-2ζ94000000000    orthogonal faithful
ρ264-40-20020098+2ζ997+2ζ9295+2ζ94000-2ζ97-2ζ92-2ζ95-2ζ94-2ζ98-2ζ9000000000    orthogonal faithful
ρ274-40-20020097+2ζ9295+2ζ9498+2ζ9000-2ζ95-2ζ94-2ζ98-2ζ9-2ζ97-2ζ92000000000    orthogonal faithful

Smallest permutation representation of Q82D9
On 72 points
Generators in S72
(1 32 14 23)(2 33 15 24)(3 34 16 25)(4 35 17 26)(5 36 18 27)(6 28 10 19)(7 29 11 20)(8 30 12 21)(9 31 13 22)(37 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 72)
(1 50 14 41)(2 51 15 42)(3 52 16 43)(4 53 17 44)(5 54 18 45)(6 46 10 37)(7 47 11 38)(8 48 12 39)(9 49 13 40)(19 64 28 55)(20 65 29 56)(21 66 30 57)(22 67 31 58)(23 68 32 59)(24 69 33 60)(25 70 34 61)(26 71 35 62)(27 72 36 63)
(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 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(27 36)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 63)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 72)

G:=sub<Sym(72)| (1,32,14,23)(2,33,15,24)(3,34,16,25)(4,35,17,26)(5,36,18,27)(6,28,10,19)(7,29,11,20)(8,30,12,21)(9,31,13,22)(37,55,46,64)(38,56,47,65)(39,57,48,66)(40,58,49,67)(41,59,50,68)(42,60,51,69)(43,61,52,70)(44,62,53,71)(45,63,54,72), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63), (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)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,72)>;

G:=Group( (1,32,14,23)(2,33,15,24)(3,34,16,25)(4,35,17,26)(5,36,18,27)(6,28,10,19)(7,29,11,20)(8,30,12,21)(9,31,13,22)(37,55,46,64)(38,56,47,65)(39,57,48,66)(40,58,49,67)(41,59,50,68)(42,60,51,69)(43,61,52,70)(44,62,53,71)(45,63,54,72), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63), (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)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,72) );

G=PermutationGroup([(1,32,14,23),(2,33,15,24),(3,34,16,25),(4,35,17,26),(5,36,18,27),(6,28,10,19),(7,29,11,20),(8,30,12,21),(9,31,13,22),(37,55,46,64),(38,56,47,65),(39,57,48,66),(40,58,49,67),(41,59,50,68),(42,60,51,69),(43,61,52,70),(44,62,53,71),(45,63,54,72)], [(1,50,14,41),(2,51,15,42),(3,52,16,43),(4,53,17,44),(5,54,18,45),(6,46,10,37),(7,47,11,38),(8,48,12,39),(9,49,13,40),(19,64,28,55),(20,65,29,56),(21,66,30,57),(22,67,31,58),(23,68,32,59),(24,69,33,60),(25,70,34,61),(26,71,35,62),(27,72,36,63)], [(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),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(27,36),(37,62),(38,61),(39,60),(40,59),(41,58),(42,57),(43,56),(44,55),(45,63),(46,71),(47,70),(48,69),(49,68),(50,67),(51,66),(52,65),(53,64),(54,72)])

Q82D9 is a maximal subgroup of
SD16×D9  D72⋊C2  Q16⋊D9  D725C2  C36.C23  D4⋊D18  D4.9D18  Q82D27  Q8⋊D27  Dic6⋊D9  C18.D12  D36.C6  C36.20D6  C18.6S4
Q82D9 is a maximal quotient of
C4.Dic18  C18.D8  Q82Dic9  Q82D27  Dic6⋊D9  C18.D12  C36.20D6

Matrix representation of Q82D9 in GL4(𝔽73) generated by

1000
0100
00171
00172
,
1000
0100
006112
006712
,
704500
284200
0010
0001
,
284200
704500
0010
00172
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,71,72],[1,0,0,0,0,1,0,0,0,0,61,67,0,0,12,12],[70,28,0,0,45,42,0,0,0,0,1,0,0,0,0,1],[28,70,0,0,42,45,0,0,0,0,1,1,0,0,0,72] >;

Q82D9 in GAP, Magma, Sage, TeX

Q_8\rtimes_2D_9
% in TeX

G:=Group("Q8:2D9");
// GroupNames label

G:=SmallGroup(144,18);
// by ID

G=gap.SmallGroup(144,18);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,55,218,116,50,2404,208,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^9=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of Q82D9 in TeX
Character table of Q82D9 in TeX

׿
×
𝔽