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G = Q8×D9order 144 = 24·32

Direct product of Q8 and D9

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

Aliases: Q8×D9, C12.7D6, C4.6D18, Dic184C2, C36.6C22, C18.7C23, D18.5C22, Dic9.4C22, C92(C2×Q8), C3.(S3×Q8), (Q8×C9)⋊2C2, (C4×D9).1C2, (C3×Q8).7S3, C2.8(C22×D9), C6.25(C22×S3), SmallGroup(144,43)

Series: Derived Chief Lower central Upper central

C1C18 — Q8×D9
C1C3C9C18D18C4×D9 — Q8×D9
C9C18 — Q8×D9
C1C2Q8

Generators and relations for Q8×D9
 G = < a,b,c,d | a4=c9=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 187 in 57 conjugacy classes, 31 normal (11 characteristic)
C1, C2, C2 [×2], C3, C4 [×3], C4 [×3], C22, S3 [×2], C6, C2×C4 [×3], Q8, Q8 [×3], C9, Dic3 [×3], C12 [×3], D6, C2×Q8, D9 [×2], C18, Dic6 [×3], C4×S3 [×3], C3×Q8, Dic9 [×3], C36 [×3], D18, S3×Q8, Dic18 [×3], C4×D9 [×3], Q8×C9, Q8×D9
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D6 [×3], C2×Q8, D9, C22×S3, D18 [×3], S3×Q8, C22×D9, Q8×D9

Character table of Q8×D9

 class 12A2B2C34A4B4C4D4E4F69A9B9C12A12B12C18A18B18C36A36B36C36D36E36F36G36H36I
 size 119922221818182222444222444444444
ρ1111111111111111111111111111111    trivial
ρ211-1-11111-1-1-11111111111111111111    linear of order 2
ρ311-1-11-1-11-1111111-11-1111-1111-1-1-1-1-1    linear of order 2
ρ411111-1-111-1-11111-11-1111-1111-1-1-1-1-1    linear of order 2
ρ5111111-1-1-1-111111-1-11111-1-1-1-1-1111-1    linear of order 2
ρ611-1-111-1-111-11111-1-11111-1-1-1-1-1111-1    linear of order 2
ρ711-1-11-11-11-1111111-1-11111-1-1-11-1-1-11    linear of order 2
ρ811111-11-1-11-111111-1-11111-1-1-11-1-1-11    linear of order 2
ρ922002-22-20002-1-1-12-2-2-1-1-1-1111-1111-1    orthogonal lifted from D6
ρ10220022220002-1-1-1222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1122002-2-220002-1-1-1-22-2-1-1-11-1-1-111111    orthogonal lifted from D6
ρ12220022-2-20002-1-1-1-2-22-1-1-111111-1-1-11    orthogonal lifted from D6
ρ132200-1222000-1ζ9594ζ9792ζ989-1-1-1ζ989ζ9594ζ9792ζ9594ζ9594ζ989ζ9792ζ989ζ9594ζ989ζ9792ζ9792    orthogonal lifted from D9
ρ142200-1-22-2000-1ζ9594ζ9792ζ989-111ζ989ζ9594ζ9792ζ959495949899792ζ98995949899792ζ9792    orthogonal lifted from D18
ρ152200-12-2-2000-1ζ989ζ9594ζ979211-1ζ9792ζ989ζ9594989989979295949792ζ989ζ9792ζ95949594    orthogonal lifted from D18
ρ162200-1-22-2000-1ζ989ζ9594ζ9792-111ζ9792ζ989ζ9594ζ98998997929594ζ979298997929594ζ9594    orthogonal lifted from D18
ρ172200-1222000-1ζ9792ζ989ζ9594-1-1-1ζ9594ζ9792ζ989ζ9792ζ9792ζ9594ζ989ζ9594ζ9792ζ9594ζ989ζ989    orthogonal lifted from D9
ρ182200-12-2-2000-1ζ9594ζ9792ζ98911-1ζ989ζ9594ζ9792959495949899792989ζ9594ζ989ζ97929792    orthogonal lifted from D18
ρ192200-1-2-22000-1ζ989ζ9594ζ97921-11ζ9792ζ989ζ9594989ζ989ζ9792ζ95949792989979295949594    orthogonal lifted from D18
ρ202200-1222000-1ζ989ζ9594ζ9792-1-1-1ζ9792ζ989ζ9594ζ989ζ989ζ9792ζ9594ζ9792ζ989ζ9792ζ9594ζ9594    orthogonal lifted from D9
ρ212200-1-22-2000-1ζ9792ζ989ζ9594-111ζ9594ζ9792ζ989ζ979297929594989ζ959497929594989ζ989    orthogonal lifted from D18
ρ222200-1-2-22000-1ζ9792ζ989ζ95941-11ζ9594ζ9792ζ9899792ζ9792ζ9594ζ989959497929594989989    orthogonal lifted from D18
ρ232200-1-2-22000-1ζ9594ζ9792ζ9891-11ζ989ζ9594ζ97929594ζ9594ζ989ζ9792989959498997929792    orthogonal lifted from D18
ρ242200-12-2-2000-1ζ9792ζ989ζ959411-1ζ9594ζ9792ζ9899792979295949899594ζ9792ζ9594ζ989989    orthogonal lifted from D18
ρ252-22-22000000-2222000-2-2-2000000000    symplectic lifted from Q8, Schur index 2
ρ262-2-222000000-2222000-2-2-2000000000    symplectic lifted from Q8, Schur index 2
ρ274-4004000000-4-2-2-2000222000000000    symplectic lifted from S3×Q8, Schur index 2
ρ284-400-2000000297+2ζ9298+2ζ995+2ζ94000-2ζ95-2ζ94-2ζ97-2ζ92-2ζ98-2ζ9000000000    symplectic faithful, Schur index 2
ρ294-400-2000000298+2ζ995+2ζ9497+2ζ92000-2ζ97-2ζ92-2ζ98-2ζ9-2ζ95-2ζ94000000000    symplectic faithful, Schur index 2
ρ304-400-2000000295+2ζ9497+2ζ9298+2ζ9000-2ζ98-2ζ9-2ζ95-2ζ94-2ζ97-2ζ92000000000    symplectic faithful, Schur index 2

Smallest permutation representation of Q8×D9
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 13)(2 12)(3 11)(4 10)(5 18)(6 17)(7 16)(8 15)(9 14)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(27 36)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(45 54)(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), (1,13)(2,12)(3,11)(4,10)(5,18)(6,17)(7,16)(8,15)(9,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(45,54)(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), (1,13)(2,12)(3,11)(4,10)(5,18)(6,17)(7,16)(8,15)(9,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(45,54)(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)], [(1,13),(2,12),(3,11),(4,10),(5,18),(6,17),(7,16),(8,15),(9,14),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(27,36),(37,53),(38,52),(39,51),(40,50),(41,49),(42,48),(43,47),(44,46),(45,54),(55,71),(56,70),(57,69),(58,68),(59,67),(60,66),(61,65),(62,64),(63,72)])

Q8×D9 is a maximal subgroup of
SD16⋊D9  Q16⋊D9  Q8.15D18  D4.10D18  D18.A4  Dic18⋊S3
Q8×D9 is a maximal quotient of
Dic93Q8  C36⋊Q8  Dic9.Q8  D18⋊Q8  D182Q8  Dic9⋊Q8  D183Q8  Dic18⋊S3

Matrix representation of Q8×D9 in GL4(𝔽37) generated by

1000
0100
0001
00360
,
1000
0100
002332
003214
,
171100
26600
0010
0001
,
171100
312000
00360
00036
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,0,36,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,23,32,0,0,32,14],[17,26,0,0,11,6,0,0,0,0,1,0,0,0,0,1],[17,31,0,0,11,20,0,0,0,0,36,0,0,0,0,36] >;

Q8×D9 in GAP, Magma, Sage, TeX

Q_8\times D_9
% in TeX

G:=Group("Q8xD9");
// GroupNames label

G:=SmallGroup(144,43);
// by ID

G=gap.SmallGroup(144,43);
# by ID

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

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

Export

Character table of Q8×D9 in TeX

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