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G = C9⋊D4order 72 = 23·32

The semidirect product of C9 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C92D4, Dic9⋊C2, D182C2, C6.10D6, C2.5D18, C222D9, C18.5C22, (C2×C18)⋊2C2, (C2×C6).3S3, C3.(C3⋊D4), SmallGroup(72,8)

Series: Derived Chief Lower central Upper central

C1C18 — C9⋊D4
C1C3C9C18D18 — C9⋊D4
C9C18 — C9⋊D4
C1C2C22

Generators and relations for C9⋊D4
 G = < a,b,c | a9=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
18C2
9C4
9C22
2C6
6S3
9D4
3Dic3
3D6
2D9
2C18
3C3⋊D4

Character table of C9⋊D4

 class 12A2B2C346A6B6C9A9B9C18A18B18C18D18E18F18G18H18I
 size 11218218222222222222222
ρ1111111111111111111111    trivial
ρ211-1-111-11-11111-1-1-1-1-1-111    linear of order 2
ρ3111-11-1111111111111111    linear of order 2
ρ411-111-1-11-11111-1-1-1-1-1-111    linear of order 2
ρ5222020222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ62-200200-20222-2000000-2-2    orthogonal lifted from D4
ρ722-2020-22-2-1-1-1-1111111-1-1    orthogonal lifted from D6
ρ822-20-101-11ζ9594ζ9792ζ989ζ9899792959498995949792989ζ9792ζ9594    orthogonal lifted from D18
ρ922-20-101-11ζ989ζ9594ζ9792ζ97929594989979298995949792ζ9594ζ989    orthogonal lifted from D18
ρ102220-10-1-1-1ζ9792ζ989ζ9594ζ9594ζ989ζ9792ζ9594ζ9792ζ989ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ112220-10-1-1-1ζ9594ζ9792ζ989ζ989ζ9792ζ9594ζ989ζ9594ζ9792ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ1222-20-101-11ζ9792ζ989ζ9594ζ95949899792959497929899594ζ989ζ9792    orthogonal lifted from D18
ρ132220-10-1-1-1ζ989ζ9594ζ9792ζ9792ζ9594ζ989ζ9792ζ989ζ9594ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ142-200200-20-1-1-11--3--3--3-3-3-311    complex lifted from C3⋊D4
ρ152-200-10--31-3ζ9792ζ989ζ95949594ζ9899792ζ9594ζ979298995949899792    complex faithful
ρ162-200200-20-1-1-11-3-3-3--3--3--311    complex lifted from C3⋊D4
ρ172-200-10--31-3ζ9594ζ9792ζ9899899792ζ9594ζ9899594ζ979298997929594    complex faithful
ρ182-200-10--31-3ζ989ζ9594ζ97929792ζ9594ζ98997929899594ζ97929594989    complex faithful
ρ192-200-10-31--3ζ9792ζ989ζ95949594989ζ979295949792ζ989ζ95949899792    complex faithful
ρ202-200-10-31--3ζ9594ζ9792ζ989989ζ97929594989ζ95949792ζ98997929594    complex faithful
ρ212-200-10-31--3ζ989ζ9594ζ979297929594989ζ9792ζ989ζ959497929594989    complex faithful

Smallest permutation representation of C9⋊D4
On 36 points
Generators in S36
(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)
(1 20 11 29)(2 19 12 28)(3 27 13 36)(4 26 14 35)(5 25 15 34)(6 24 16 33)(7 23 17 32)(8 22 18 31)(9 21 10 30)
(2 9)(3 8)(4 7)(5 6)(10 12)(13 18)(14 17)(15 16)(19 30)(20 29)(21 28)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)

G:=sub<Sym(36)| (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), (1,20,11,29)(2,19,12,28)(3,27,13,36)(4,26,14,35)(5,25,15,34)(6,24,16,33)(7,23,17,32)(8,22,18,31)(9,21,10,30), (2,9)(3,8)(4,7)(5,6)(10,12)(13,18)(14,17)(15,16)(19,30)(20,29)(21,28)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)>;

G:=Group( (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), (1,20,11,29)(2,19,12,28)(3,27,13,36)(4,26,14,35)(5,25,15,34)(6,24,16,33)(7,23,17,32)(8,22,18,31)(9,21,10,30), (2,9)(3,8)(4,7)(5,6)(10,12)(13,18)(14,17)(15,16)(19,30)(20,29)(21,28)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31) );

G=PermutationGroup([(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)], [(1,20,11,29),(2,19,12,28),(3,27,13,36),(4,26,14,35),(5,25,15,34),(6,24,16,33),(7,23,17,32),(8,22,18,31),(9,21,10,30)], [(2,9),(3,8),(4,7),(5,6),(10,12),(13,18),(14,17),(15,16),(19,30),(20,29),(21,28),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31)])

C9⋊D4 is a maximal subgroup of
D365C2  D4×D9  D42D9  C27⋊D4  C9.S4  D6⋊D9  C9⋊D12  Dic9⋊C6  C6.D18  C9⋊S4  Q8.D18  C23.D18  C45⋊D4  C9⋊D20  C457D4
C9⋊D4 is a maximal quotient of
Dic9⋊C4  D18⋊C4  D4.D9  D4⋊D9  C9⋊Q16  Q82D9  C18.D4  C27⋊D4  D6⋊D9  C9⋊D12  C6.D18  C23.D18  C45⋊D4  C9⋊D20  C457D4

Matrix representation of C9⋊D4 in GL2(𝔽19) generated by

815
1514
,
124
167
,
1811
01
G:=sub<GL(2,GF(19))| [8,15,15,14],[12,16,4,7],[18,0,11,1] >;

C9⋊D4 in GAP, Magma, Sage, TeX

C_9\rtimes D_4
% in TeX

G:=Group("C9:D4");
// GroupNames label

G:=SmallGroup(72,8);
// by ID

G=gap.SmallGroup(72,8);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,61,803,138,1204]);
// Polycyclic

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

Export

Subgroup lattice of C9⋊D4 in TeX
Character table of C9⋊D4 in TeX

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