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G = D4⋊D9order 144 = 24·32

The semidirect product of D4 and D9 acting via D9/C9=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4⋊D9, C92D8, D362C2, C18.8D4, C4.2D18, C12.2D6, C36.2C22, C9⋊C82C2, (D4×C9)⋊1C2, C3.(D4⋊S3), (C3×D4).2S3, C2.5(C9⋊D4), C6.15(C3⋊D4), SmallGroup(144,16)

Series: Derived Chief Lower central Upper central

C1C36 — D4⋊D9
C1C3C9C18C36D36 — D4⋊D9
C9C18C36 — D4⋊D9
C1C2C4D4

Generators and relations for D4⋊D9
 G = < a,b,c,d | a4=b2=c9=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >

4C2
36C2
2C22
18C22
4C6
12S3
9C8
9D4
2C2×C6
6D6
4C18
4D9
9D8
3C3⋊C8
3D12
2D18
2C2×C18
3D4⋊S3

Character table of D4⋊D9

 class 12A2B2C346A6B6C8A8B9A9B9C1218A18B18C18D18E18F18G18H18I36A36B36C
 size 114362224418182224222444444444
ρ1111111111111111111111111111    trivial
ρ2111-111111-1-11111111111111111    linear of order 2
ρ311-1-1111-1-1111111111-1-1-1-1-1-1111    linear of order 2
ρ411-11111-1-1-1-11111111-1-1-1-1-1-1111    linear of order 2
ρ522-20222-2-200-1-1-12-1-1-1111111-1-1-1    orthogonal lifted from D6
ρ622002-220000222-2222000000-2-2-2    orthogonal lifted from D4
ρ722202222200-1-1-12-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ82220-12-1-1-100ζ9594ζ989ζ9792-1ζ9594ζ989ζ9792ζ9792ζ9594ζ9792ζ989ζ9594ζ989ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ922-20-12-11100ζ9792ζ9594ζ989-1ζ9792ζ9594ζ9899899792989959497929594ζ9594ζ989ζ9792    orthogonal lifted from D18
ρ102-20020-2002-22220-2-2-2000000000    orthogonal lifted from D8
ρ112-20020-200-222220-2-2-2000000000    orthogonal lifted from D8
ρ122220-12-1-1-100ζ9792ζ9594ζ989-1ζ9792ζ9594ζ989ζ989ζ9792ζ989ζ9594ζ9792ζ9594ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ1322-20-12-11100ζ9594ζ989ζ9792-1ζ9594ζ989ζ97929792959497929899594989ζ989ζ9792ζ9594    orthogonal lifted from D18
ρ1422-20-12-11100ζ989ζ9792ζ9594-1ζ989ζ9792ζ95949594989959497929899792ζ9792ζ9594ζ989    orthogonal lifted from D18
ρ152220-12-1-1-100ζ989ζ9792ζ9594-1ζ989ζ9792ζ9594ζ9594ζ989ζ9594ζ9792ζ989ζ9792ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ1622002-220000-1-1-1-2-1-1-1--3--3-3-3-3--3111    complex lifted from C3⋊D4
ρ1722002-220000-1-1-1-2-1-1-1-3-3--3--3--3-3111    complex lifted from C3⋊D4
ρ182200-1-2-1-3--300ζ989ζ9792ζ95941ζ989ζ9792ζ95949594989ζ95949792ζ989ζ979297929594989    complex lifted from C9⋊D4
ρ192200-1-2-1--3-300ζ989ζ9792ζ95941ζ989ζ9792ζ9594ζ9594ζ9899594ζ9792989979297929594989    complex lifted from C9⋊D4
ρ202200-1-2-1--3-300ζ9792ζ9594ζ9891ζ9792ζ9594ζ989ζ98997929899594ζ9792ζ959495949899792    complex lifted from C9⋊D4
ρ212200-1-2-1-3--300ζ9594ζ989ζ97921ζ9594ζ989ζ9792ζ979295949792ζ989ζ959498998997929594    complex lifted from C9⋊D4
ρ222200-1-2-1-3--300ζ9792ζ9594ζ9891ζ9792ζ9594ζ989989ζ9792ζ989ζ95949792959495949899792    complex lifted from C9⋊D4
ρ232200-1-2-1--3-300ζ9594ζ989ζ97921ζ9594ζ989ζ97929792ζ9594ζ97929899594ζ98998997929594    complex lifted from C9⋊D4
ρ244-40040-40000-2-2-20222000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ254-400-202000098+2ζ997+2ζ9295+2ζ940-2ζ98-2ζ9-2ζ97-2ζ92-2ζ95-2ζ94000000000    orthogonal faithful, Schur index 2
ρ264-400-202000095+2ζ9498+2ζ997+2ζ920-2ζ95-2ζ94-2ζ98-2ζ9-2ζ97-2ζ92000000000    orthogonal faithful, Schur index 2
ρ274-400-202000097+2ζ9295+2ζ9498+2ζ90-2ζ97-2ζ92-2ζ95-2ζ94-2ζ98-2ζ9000000000    orthogonal faithful, Schur index 2

Smallest permutation representation of D4⋊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 68)(2 69)(3 70)(4 71)(5 72)(6 64)(7 65)(8 66)(9 67)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(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 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), (1,68)(2,69)(3,70)(4,71)(5,72)(6,64)(7,65)(8,66)(9,67)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(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)(37,62)(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), (1,68)(2,69)(3,70)(4,71)(5,72)(6,64)(7,65)(8,66)(9,67)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,61)(17,62)(18,63)(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)(37,62)(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)], [(1,68),(2,69),(3,70),(4,71),(5,72),(6,64),(7,65),(8,66),(9,67),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,61),(17,62),(18,63),(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),(37,62),(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)])

D4⋊D9 is a maximal subgroup of
D8×D9  D8⋊D9  D72⋊C2  SD163D9  D366C22  D4⋊D18  D4.9D18  D4⋊D27  D36⋊S3  C9⋊D24  D36⋊C6  C36.18D6
D4⋊D9 is a maximal quotient of
C36.Q8  C18.D8  C9⋊D16  D8.D9  C9⋊SD32  C9⋊Q32  D4⋊Dic9  D4⋊D27  D36⋊S3  C9⋊D24  C36.18D6

Matrix representation of D4⋊D9 in GL4(𝔽73) generated by

1000
0100
0013
004872
,
1000
0100
00048
00350
,
453100
42300
0010
0001
,
42300
453100
0010
004872
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,48,0,0,3,72],[1,0,0,0,0,1,0,0,0,0,0,35,0,0,48,0],[45,42,0,0,31,3,0,0,0,0,1,0,0,0,0,1],[42,45,0,0,3,31,0,0,0,0,1,48,0,0,0,72] >;

D4⋊D9 in GAP, Magma, Sage, TeX

D_4\rtimes D_9
% in TeX

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

G:=SmallGroup(144,16);
// by ID

G=gap.SmallGroup(144,16);
# by ID

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

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

Export

Subgroup lattice of D4⋊D9 in TeX
Character table of D4⋊D9 in TeX

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