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G = Q83D9order 144 = 24·32

The semidirect product of Q8 and D9 acting through Inn(Q8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q83D9, D364C2, C12.8D6, C4.7D18, C36.7C22, C18.8C23, D18.3C22, Dic9.5C22, (C4×D9)⋊3C2, C93(C4○D4), (Q8×C9)⋊3C2, (C3×Q8).8S3, C3.(Q83S3), C2.9(C22×D9), C6.26(C22×S3), SmallGroup(144,44)

Series: Derived Chief Lower central Upper central

C1C18 — Q83D9
C1C3C9C18D18C4×D9 — Q83D9
C9C18 — Q83D9
C1C2Q8

Generators and relations for Q83D9
 G = < a,b,c,d | a4=c9=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

Subgroups: 239 in 60 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2 [×3], C3, C4 [×3], C4, C22 [×3], S3 [×3], C6, C2×C4 [×3], D4 [×3], Q8, C9, Dic3, C12 [×3], D6 [×3], C4○D4, D9 [×3], C18, C4×S3 [×3], D12 [×3], C3×Q8, Dic9, C36 [×3], D18 [×3], Q83S3, C4×D9 [×3], D36 [×3], Q8×C9, Q83D9
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], C4○D4, D9, C22×S3, D18 [×3], Q83S3, C22×D9, Q83D9

Character table of Q83D9

 class 12A2B2C2D34A4B4C4D4E69A9B9C12A12B12C18A18B18C36A36B36C36D36E36F36G36H36I
 size 111818182222992222444222444444444
ρ1111111111111111111111111111111    trivial
ρ2111-111-1-11-1-11111-1-11111-1111-1-1-1-1-1    linear of order 2
ρ3111-1-11-11-11111111-1-11111-1-1-1-1-111-1    linear of order 2
ρ41111-111-1-1-1-11111-11-1111-1-1-1-111-1-11    linear of order 2
ρ511-1-1-11111-1-11111111111111111111    linear of order 2
ρ611-11-11-1-11111111-1-11111-1111-1-1-1-1-1    linear of order 2
ρ711-1111-11-1-1-111111-1-11111-1-1-1-1-111-1    linear of order 2
ρ811-1-1111-1-1111111-11-1111-1-1-1-111-1-11    linear of order 2
ρ9220002222002-1-1-1222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ10220002-2-22002-1-1-1-2-22-1-1-11-1-1-111111    orthogonal lifted from D6
ρ11220002-22-2002-1-1-12-2-2-1-1-1-111111-1-11    orthogonal lifted from D6
ρ122200022-2-2002-1-1-1-22-2-1-1-11111-1-111-1    orthogonal lifted from D6
ρ1322000-1-22-200-1ζ9594ζ9792ζ989-111ζ9594ζ989ζ9792ζ9594989979295949594989ζ989ζ97929792    orthogonal lifted from D18
ρ1422000-1-2-2200-1ζ9594ζ9792ζ98911-1ζ9594ζ989ζ97929594ζ989ζ9792ζ9594959498998997929792    orthogonal lifted from D18
ρ1522000-1-22-200-1ζ989ζ9594ζ9792-111ζ989ζ9792ζ9594ζ989979295949899899792ζ9792ζ95949594    orthogonal lifted from D18
ρ1622000-1-22-200-1ζ9792ζ989ζ9594-111ζ9792ζ9594ζ989ζ97929594989979297929594ζ9594ζ989989    orthogonal lifted from D18
ρ1722000-122200-1ζ989ζ9594ζ9792-1-1-1ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ989ζ9792ζ9792ζ9594ζ9594    orthogonal lifted from D9
ρ1822000-12-2-200-1ζ9792ζ989ζ95941-11ζ9792ζ9594ζ989979295949899792ζ9792ζ95949594989ζ989    orthogonal lifted from D18
ρ1922000-1-2-2200-1ζ9792ζ989ζ959411-1ζ9792ζ9594ζ9899792ζ9594ζ989ζ9792979295949594989989    orthogonal lifted from D18
ρ2022000-1-2-2200-1ζ989ζ9594ζ979211-1ζ989ζ9792ζ9594989ζ9792ζ9594ζ9899899792979295949594    orthogonal lifted from D18
ρ2122000-122200-1ζ9792ζ989ζ9594-1-1-1ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9792ζ9594ζ9594ζ989ζ989    orthogonal lifted from D9
ρ2222000-12-2-200-1ζ989ζ9594ζ97921-11ζ989ζ9792ζ959498997929594989ζ989ζ979297929594ζ9594    orthogonal lifted from D18
ρ2322000-12-2-200-1ζ9594ζ9792ζ9891-11ζ9594ζ989ζ9792959498997929594ζ9594ζ9899899792ζ9792    orthogonal lifted from D18
ρ2422000-122200-1ζ9594ζ9792ζ989-1-1-1ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ9594ζ989ζ989ζ9792ζ9792    orthogonal lifted from D9
ρ252-20002000-2i2i-2222000-2-2-2000000000    complex lifted from C4○D4
ρ262-200020002i-2i-2222000-2-2-2000000000    complex lifted from C4○D4
ρ274-4000400000-4-2-2-2000222000000000    orthogonal lifted from Q83S3, Schur index 2
ρ284-4000-200000295+2ζ9497+2ζ9298+2ζ9000-2ζ95-2ζ94-2ζ98-2ζ9-2ζ97-2ζ92000000000    orthogonal faithful, Schur index 2
ρ294-4000-200000297+2ζ9298+2ζ995+2ζ94000-2ζ97-2ζ92-2ζ95-2ζ94-2ζ98-2ζ9000000000    orthogonal faithful, Schur index 2
ρ304-4000-200000298+2ζ995+2ζ9497+2ζ92000-2ζ98-2ζ9-2ζ97-2ζ92-2ζ95-2ζ94000000000    orthogonal faithful, Schur index 2

Smallest permutation representation of Q83D9
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 44)(38 43)(39 42)(40 41)(46 53)(47 52)(48 51)(49 50)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(63 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)(37,44)(38,43)(39,42)(40,41)(46,53)(47,52)(48,51)(49,50)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(63,72)>;

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)(37,44)(38,43)(39,42)(40,41)(46,53)(47,52)(48,51)(49,50)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(63,72) );

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),(37,44),(38,43),(39,42),(40,41),(46,53),(47,52),(48,51),(49,50),(55,71),(56,70),(57,69),(58,68),(59,67),(60,66),(61,65),(62,64),(63,72)])

Q83D9 is a maximal subgroup of
D72⋊C2  SD163D9  Q16⋊D9  D725C2  Q8.15D18  C4○D4×D9  D48D18  Q83D27  Dic9.A4  Dic9.2A4  D18.D6  Dic65D9  D363C6  C36.29D6
Q83D9 is a maximal quotient of
C36.3Q8  C4⋊C47D9  D36⋊C4  D18.D4  C4⋊D36  C4⋊C4⋊D9  Q8×Dic9  D183Q8  C36.23D4  Q83D27  D18.D6  Dic65D9  C36.29D6

Matrix representation of Q83D9 in GL4(𝔽37) generated by

1000
0100
003635
0011
,
36000
03600
00310
0066
,
20600
312600
0010
0001
,
36000
36100
0010
003636
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,36,1,0,0,35,1],[36,0,0,0,0,36,0,0,0,0,31,6,0,0,0,6],[20,31,0,0,6,26,0,0,0,0,1,0,0,0,0,1],[36,36,0,0,0,1,0,0,0,0,1,36,0,0,0,36] >;

Q83D9 in GAP, Magma, Sage, TeX

Q_8\rtimes_3D_9
% in TeX

G:=Group("Q8:3D9");
// GroupNames label

G:=SmallGroup(144,44);
// by ID

G=gap.SmallGroup(144,44);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,116,50,2404,208,3461]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

Character table of Q83D9 in TeX

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