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G = D4.D9order 144 = 24·32

The non-split extension by D4 of D9 acting via D9/C9=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.D9, C92SD16, C4.1D18, C18.7D4, C12.1D6, Dic182C2, C36.1C22, C9⋊C81C2, (C3×D4).1S3, (D4×C9).1C2, C3.(D4.S3), C2.4(C9⋊D4), C6.14(C3⋊D4), SmallGroup(144,15)

Series: Derived Chief Lower central Upper central

C1C36 — D4.D9
C1C3C9C18C36Dic18 — D4.D9
C9C18C36 — D4.D9
C1C2C4D4

Generators and relations for D4.D9
 G = < a,b,c,d | a4=b2=c9=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

4C2
2C22
18C4
4C6
9C8
9Q8
2C2×C6
6Dic3
4C18
9SD16
3C3⋊C8
3Dic6
2Dic9
2C2×C18
3D4.S3

Character table of D4.D9

 class 12A2B34A4B6A6B6C8A8B9A9B9C1218A18B18C18D18E18F18G18H18I36A36B36C
 size 114223624418182224222444444444
ρ1111111111111111111111111111    trivial
ρ211-111-11-1-1111111111-1-1-1-1-1-1111    linear of order 2
ρ311-11111-1-1-1-11111111-1-1-1-1-1-1111    linear of order 2
ρ411111-1111-1-11111111111111111    linear of order 2
ρ522-22202-2-200-1-1-12-1-1-1111111-1-1-1    orthogonal lifted from D6
ρ622222022200-1-1-12-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ72202-2020000222-2222000000-2-2-2    orthogonal lifted from D4
ρ822-2-120-11100ζ9594ζ989ζ9792-1ζ9594ζ989ζ97929792959497929899594989ζ989ζ9792ζ9594    orthogonal lifted from D18
ρ9222-120-1-1-100ζ9594ζ989ζ9792-1ζ9594ζ989ζ9792ζ9792ζ9594ζ9792ζ989ζ9594ζ989ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ1022-2-120-11100ζ989ζ9792ζ9594-1ζ989ζ9792ζ95949594989959497929899792ζ9792ζ9594ζ989    orthogonal lifted from D18
ρ11222-120-1-1-100ζ9792ζ9594ζ989-1ζ9792ζ9594ζ989ζ989ζ9792ζ989ζ9594ζ9792ζ9594ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ12222-120-1-1-100ζ989ζ9792ζ9594-1ζ989ζ9792ζ9594ζ9594ζ989ζ9594ζ9792ζ989ζ9792ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ1322-2-120-11100ζ9792ζ9594ζ989-1ζ9792ζ9594ζ9899899792989959497929594ζ9594ζ989ζ9792    orthogonal lifted from D18
ρ142-20200-200-2--22220-2-2-2000000000    complex lifted from SD16
ρ152-20200-200--2-22220-2-2-2000000000    complex lifted from SD16
ρ162202-2020000-1-1-1-2-1-1-1--3--3-3-3-3--3111    complex lifted from C3⋊D4
ρ172202-2020000-1-1-1-2-1-1-1-3-3--3--3--3-3111    complex lifted from C3⋊D4
ρ18220-1-20-1-3--300ζ9792ζ9594ζ9891ζ9792ζ9594ζ989989ζ9792ζ989ζ95949792959495949899792    complex lifted from C9⋊D4
ρ19220-1-20-1--3-300ζ989ζ9792ζ95941ζ989ζ9792ζ9594ζ9594ζ9899594ζ9792989979297929594989    complex lifted from C9⋊D4
ρ20220-1-20-1-3--300ζ9594ζ989ζ97921ζ9594ζ989ζ9792ζ979295949792ζ989ζ959498998997929594    complex lifted from C9⋊D4
ρ21220-1-20-1-3--300ζ989ζ9792ζ95941ζ989ζ9792ζ95949594989ζ95949792ζ989ζ979297929594989    complex lifted from C9⋊D4
ρ22220-1-20-1--3-300ζ9792ζ9594ζ9891ζ9792ζ9594ζ989ζ98997929899594ζ9792ζ959495949899792    complex lifted from C9⋊D4
ρ23220-1-20-1--3-300ζ9594ζ989ζ97921ζ9594ζ989ζ97929792ζ9594ζ97929899594ζ98998997929594    complex lifted from C9⋊D4
ρ244-40400-40000-2-2-20222000000000    symplectic lifted from D4.S3, Schur index 2
ρ254-40-2002000097+2ζ9295+2ζ9498+2ζ90-2ζ97-2ζ92-2ζ95-2ζ94-2ζ98-2ζ9000000000    symplectic faithful, Schur index 2
ρ264-40-2002000095+2ζ9498+2ζ997+2ζ920-2ζ95-2ζ94-2ζ98-2ζ9-2ζ97-2ζ92000000000    symplectic faithful, Schur index 2
ρ274-40-2002000098+2ζ997+2ζ9295+2ζ940-2ζ98-2ζ9-2ζ97-2ζ92-2ζ95-2ζ94000000000    symplectic 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 23)(2 24)(3 25)(4 26)(5 27)(6 19)(7 20)(8 21)(9 22)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 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 48 14 39)(2 47 15 38)(3 46 16 37)(4 54 17 45)(5 53 18 44)(6 52 10 43)(7 51 11 42)(8 50 12 41)(9 49 13 40)(19 70 28 61)(20 69 29 60)(21 68 30 59)(22 67 31 58)(23 66 32 57)(24 65 33 56)(25 64 34 55)(26 72 35 63)(27 71 36 62)

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

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

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

D4.D9 is a maximal subgroup of
D8⋊D9  D83D9  SD16×D9  SD16⋊D9  D366C22  D4.D18  D4.9D18  D4.D27  D12.D9  C36.D6  Dic18⋊C6  C36.17D6
D4.D9 is a maximal quotient of
C4.Dic18  C18.Q16  D4⋊Dic9  D4.D27  D12.D9  C36.D6  C36.17D6

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

72000
07200
007270
00251
,
72000
0100
0010
004872
,
32000
01600
0010
0001
,
01600
32000
00055
00690
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,25,0,0,70,1],[72,0,0,0,0,1,0,0,0,0,1,48,0,0,0,72],[32,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,32,0,0,16,0,0,0,0,0,0,69,0,0,55,0] >;

D4.D9 in GAP, Magma, Sage, TeX

D_4.D_9
% in TeX

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

G:=SmallGroup(144,15);
// by ID

G=gap.SmallGroup(144,15);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,218,116,50,2404,208,3461]);
// Polycyclic

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

Export

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

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