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G = D42D9order 144 = 24·32

The semidirect product of D4 and D9 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D42D9, C12.6D6, C4.5D18, Dic183C2, C36.5C22, C18.6C23, C22.1D18, D18.2C22, Dic9.3C22, (D4×C9)⋊3C2, (C4×D9)⋊2C2, C92(C4○D4), C9⋊D42C2, (C2×C6).3D6, (C3×D4).4S3, (C2×C18).C22, C3.(D42S3), (C2×Dic9)⋊3C2, C2.7(C22×D9), C6.24(C22×S3), SmallGroup(144,42)

Series: Derived Chief Lower central Upper central

C1C18 — D42D9
C1C3C9C18D18C4×D9 — D42D9
C9C18 — D42D9
C1C2D4

Generators and relations for D42D9
 G = < a,b,c,d | a4=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 199 in 60 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×2], C22, S3, C6, C6 [×2], C2×C4 [×3], D4, D4 [×2], Q8, C9, Dic3 [×3], C12, D6, C2×C6 [×2], C4○D4, D9, C18, C18 [×2], Dic6, C4×S3, C2×Dic3 [×2], C3⋊D4 [×2], C3×D4, Dic9, Dic9 [×2], C36, D18, C2×C18 [×2], D42S3, Dic18, C4×D9, C2×Dic9 [×2], C9⋊D4 [×2], D4×C9, D42D9
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], C4○D4, D9, C22×S3, D18 [×3], D42S3, C22×D9, D42D9

Character table of D42D9

 class 12A2B2C2D34A4B4C4D4E6A6B6C9A9B9C1218A18B18C18D18E18F18G18H18I36A36B36C
 size 112218229918182442224222444444444
ρ1111111111111111111111111111111    trivial
ρ2111-111-1-1-11-111-1111-1111-1111-1-1-1-1-1    linear of order 2
ρ31111-111-1-1-1-11111111111111111111    linear of order 2
ρ4111-1-11-111-1111-1111-1111-1111-1-1-1-1-1    linear of order 2
ρ511-1111-1-1-1-111-11111-11111-1-1-111-1-1-1    linear of order 2
ρ611-1-111111-1-11-1-11111111-1-1-1-1-1-1111    linear of order 2
ρ711-11-11-1111-11-11111-11111-1-1-111-1-1-1    linear of order 2
ρ811-1-1-111-1-1111-1-11111111-1-1-1-1-1-1111    linear of order 2
ρ922-2202-200002-22-1-1-1-2-1-1-1-1111-1-1111    orthogonal lifted from D6
ρ1022-2-202200002-2-2-1-1-12-1-1-1111111-1-1-1    orthogonal lifted from D6
ρ1122220220000222-1-1-12-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ12222-202-2000022-2-1-1-1-2-1-1-11-1-1-111111    orthogonal lifted from D6
ρ1322220-120000-1-1-1ζ989ζ9594ζ9792-1ζ9792ζ989ζ9594ζ9792ζ9792ζ9594ζ989ζ9594ζ989ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ1422220-120000-1-1-1ζ9594ζ9792ζ989-1ζ989ζ9594ζ9792ζ989ζ989ζ9792ζ9594ζ9792ζ9594ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ1522-2-20-120000-111ζ9594ζ9792ζ989-1ζ989ζ9594ζ97929899899792959497929594ζ9792ζ9594ζ989    orthogonal lifted from D18
ρ1622-220-1-20000-11-1ζ989ζ9594ζ97921ζ9792ζ989ζ9594ζ979297929594989ζ9594ζ98995949899792    orthogonal lifted from D18
ρ17222-20-1-20000-1-11ζ9594ζ9792ζ9891ζ989ζ9594ζ9792989ζ989ζ9792ζ95949792959497929594989    orthogonal lifted from D18
ρ18222-20-1-20000-1-11ζ989ζ9594ζ97921ζ9792ζ989ζ95949792ζ9792ζ9594ζ989959498995949899792    orthogonal lifted from D18
ρ1922-2-20-120000-111ζ9792ζ989ζ9594-1ζ9594ζ9792ζ9899594959498997929899792ζ989ζ9792ζ9594    orthogonal lifted from D18
ρ2022-220-1-20000-11-1ζ9594ζ9792ζ9891ζ989ζ9594ζ9792ζ98998997929594ζ9792ζ959497929594989    orthogonal lifted from D18
ρ2122220-120000-1-1-1ζ9792ζ989ζ9594-1ζ9594ζ9792ζ989ζ9594ζ9594ζ989ζ9792ζ989ζ9792ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ2222-220-1-20000-11-1ζ9792ζ989ζ95941ζ9594ζ9792ζ989ζ959495949899792ζ989ζ979298997929594    orthogonal lifted from D18
ρ23222-20-1-20000-1-11ζ9792ζ989ζ95941ζ9594ζ9792ζ9899594ζ9594ζ989ζ9792989979298997929594    orthogonal lifted from D18
ρ2422-2-20-120000-111ζ989ζ9594ζ9792-1ζ9792ζ989ζ95949792979295949899594989ζ9594ζ989ζ9792    orthogonal lifted from D18
ρ252-2000202i-2i00-2002220-2-2-2000000000    complex lifted from C4○D4
ρ262-200020-2i2i00-2002220-2-2-2000000000    complex lifted from C4○D4
ρ274-4000400000-400-2-2-20222000000000    symplectic lifted from D42S3, Schur index 2
ρ284-4000-20000020098+2ζ995+2ζ9497+2ζ920-2ζ97-2ζ92-2ζ98-2ζ9-2ζ95-2ζ94000000000    symplectic faithful, Schur index 2
ρ294-4000-20000020097+2ζ9298+2ζ995+2ζ940-2ζ95-2ζ94-2ζ97-2ζ92-2ζ98-2ζ9000000000    symplectic faithful, Schur index 2
ρ304-4000-20000020095+2ζ9497+2ζ9298+2ζ90-2ζ98-2ζ9-2ζ95-2ζ94-2ζ97-2ζ92000000000    symplectic faithful, Schur index 2

Smallest permutation representation of D42D9
On 72 points
Generators in S72
(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 59)(2 60)(3 61)(4 62)(5 63)(6 55)(7 56)(8 57)(9 58)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)
(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)| (1,50,14,41)(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), (1,59)(2,60)(3,61)(4,62)(5,63)(6,55)(7,56)(8,57)(9,58)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54), (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( (1,50,14,41)(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), (1,59)(2,60)(3,61)(4,62)(5,63)(6,55)(7,56)(8,57)(9,58)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54), (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([(1,50,14,41),(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)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,55),(7,56),(8,57),(9,58),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54)], [(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)])

D42D9 is a maximal subgroup of
D8⋊D9  D83D9  SD16⋊D9  SD163D9  D46D18  C4○D4×D9  D4.10D18  D42D27  D125D9  D12⋊D9  Dic3.D18  D18.4D6  Dic182C6  C36.27D6
D42D9 is a maximal quotient of
C23.16D18  C222Dic18  C23.8D18  Dic94D4  C23.9D18  Dic9.D4  C22.4D36  Dic93Q8  Dic9.Q8  C36.3Q8  C4⋊C47D9  D182Q8  C4⋊C4⋊D9  D4×Dic9  C23.23D18  C36.17D4  C362D4  Dic9⋊D4  D42D27  D125D9  D12⋊D9  Dic3.D18  D18.4D6  C36.27D6

Matrix representation of D42D9 in GL4(𝔽37) generated by

1000
0100
00310
0006
,
36000
03600
0006
00310
,
20600
312600
0010
0001
,
172600
62000
0010
00036
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,31,0,0,0,0,6],[36,0,0,0,0,36,0,0,0,0,0,31,0,0,6,0],[20,31,0,0,6,26,0,0,0,0,1,0,0,0,0,1],[17,6,0,0,26,20,0,0,0,0,1,0,0,0,0,36] >;

D42D9 in GAP, Magma, Sage, TeX

D_4\rtimes_2D_9
% in TeX

G:=Group("D4:2D9");
// GroupNames label

G:=SmallGroup(144,42);
// by ID

G=gap.SmallGroup(144,42);
# by ID

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

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

Export

Character table of D42D9 in TeX

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