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G = C9⋊C8order 72 = 23·32

The semidirect product of C9 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C9⋊C8, C18.C4, C4.2D9, C2.Dic9, C36.2C2, C12.4S3, C6.1Dic3, C3.(C3⋊C8), SmallGroup(72,1)

Series: Derived Chief Lower central Upper central

C1C9 — C9⋊C8
C1C3C9C18C36 — C9⋊C8
C9 — C9⋊C8
C1C4

Generators and relations for C9⋊C8
 G = < a,b | a9=b8=1, bab-1=a-1 >

9C8
3C3⋊C8

Character table of C9⋊C8

 class 1234A4B68A8B8C8D9A9B9C12A12B18A18B18C36A36B36C36D36E36F
 size 112112999922222222222222
ρ1111111111111111111111111    trivial
ρ2111111-1-1-1-111111111111111    linear of order 2
ρ3111-1-11i-ii-i111-1-1111-1-1-1-1-1-1    linear of order 4
ρ4111-1-11-ii-ii111-1-1111-1-1-1-1-1-1    linear of order 4
ρ51-11-ii-1ζ83ζ8ζ87ζ85111-ii-1-1-1-i-iiii-i    linear of order 8
ρ61-11i-i-1ζ8ζ83ζ85ζ87111i-i-1-1-1ii-i-i-ii    linear of order 8
ρ71-11-ii-1ζ87ζ85ζ83ζ8111-ii-1-1-1-i-iiii-i    linear of order 8
ρ81-11i-i-1ζ85ζ87ζ8ζ83111i-i-1-1-1ii-i-i-ii    linear of order 8
ρ922-122-10000ζ9792ζ9594ζ989-1-1ζ9792ζ989ζ9594ζ989ζ9792ζ9792ζ9594ζ989ζ9594    orthogonal lifted from D9
ρ102222220000-1-1-122-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1122-122-10000ζ9594ζ989ζ9792-1-1ζ9594ζ9792ζ989ζ9792ζ9594ζ9594ζ989ζ9792ζ989    orthogonal lifted from D9
ρ1222-122-10000ζ989ζ9792ζ9594-1-1ζ989ζ9594ζ9792ζ9594ζ989ζ989ζ9792ζ9594ζ9792    orthogonal lifted from D9
ρ1322-1-2-2-10000ζ9792ζ9594ζ98911ζ9792ζ989ζ95949899792979295949899594    symplectic lifted from Dic9, Schur index 2
ρ1422-1-2-2-10000ζ989ζ9792ζ959411ζ989ζ9594ζ97929594989989979295949792    symplectic lifted from Dic9, Schur index 2
ρ15222-2-220000-1-1-1-2-2-1-1-1111111    symplectic lifted from Dic3, Schur index 2
ρ1622-1-2-2-10000ζ9594ζ989ζ979211ζ9594ζ9792ζ9899792959495949899792989    symplectic lifted from Dic9, Schur index 2
ρ172-222i-2i-20000-1-1-12i-2i111-i-iiii-i    complex lifted from C3⋊C8
ρ182-22-2i2i-20000-1-1-1-2i2i111ii-i-i-ii    complex lifted from C3⋊C8
ρ192-2-1-2i2i10000ζ9594ζ989ζ9792i-i95949792989ζ43ζ9743ζ92ζ43ζ9543ζ94ζ4ζ954ζ94ζ4ζ984ζ9ζ4ζ974ζ92ζ43ζ9843ζ9    complex faithful
ρ202-2-12i-2i10000ζ9792ζ9594ζ989-ii97929899594ζ4ζ984ζ9ζ4ζ974ζ92ζ43ζ9743ζ92ζ43ζ9543ζ94ζ43ζ9843ζ9ζ4ζ954ζ94    complex faithful
ρ212-2-12i-2i10000ζ989ζ9792ζ9594-ii98995949792ζ4ζ954ζ94ζ4ζ984ζ9ζ43ζ9843ζ9ζ43ζ9743ζ92ζ43ζ9543ζ94ζ4ζ974ζ92    complex faithful
ρ222-2-1-2i2i10000ζ989ζ9792ζ9594i-i98995949792ζ43ζ9543ζ94ζ43ζ9843ζ9ζ4ζ984ζ9ζ4ζ974ζ92ζ4ζ954ζ94ζ43ζ9743ζ92    complex faithful
ρ232-2-1-2i2i10000ζ9792ζ9594ζ989i-i97929899594ζ43ζ9843ζ9ζ43ζ9743ζ92ζ4ζ974ζ92ζ4ζ954ζ94ζ4ζ984ζ9ζ43ζ9543ζ94    complex faithful
ρ242-2-12i-2i10000ζ9594ζ989ζ9792-ii95949792989ζ4ζ974ζ92ζ4ζ954ζ94ζ43ζ9543ζ94ζ43ζ9843ζ9ζ43ζ9743ζ92ζ4ζ984ζ9    complex faithful

Smallest permutation representation of C9⋊C8
Regular action on 72 points
Generators in S72
(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 70 29 52 11 61 20 43)(2 69 30 51 12 60 21 42)(3 68 31 50 13 59 22 41)(4 67 32 49 14 58 23 40)(5 66 33 48 15 57 24 39)(6 65 34 47 16 56 25 38)(7 64 35 46 17 55 26 37)(8 72 36 54 18 63 27 45)(9 71 28 53 10 62 19 44)

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

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

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

C9⋊C8 is a maximal subgroup of
C8×D9  C8⋊D9  C4.Dic9  D4.D9  D4⋊D9  C9⋊Q16  Q82D9  C27⋊C8  C9⋊C24  C36.S3  C12.S4  C12.9S4  C453C8  C45⋊C8
C9⋊C8 is a maximal quotient of
C9⋊C16  C27⋊C8  C36.S3  C12.S4  C453C8  C45⋊C8

Matrix representation of C9⋊C8 in GL2(𝔽17) generated by

616
48
,
84
09
G:=sub<GL(2,GF(17))| [6,4,16,8],[8,0,4,9] >;

C9⋊C8 in GAP, Magma, Sage, TeX

C_9\rtimes C_8
% in TeX

G:=Group("C9:C8");
// GroupNames label

G:=SmallGroup(72,1);
// by ID

G=gap.SmallGroup(72,1);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,10,26,803,138,1204]);
// Polycyclic

G:=Group<a,b|a^9=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C9⋊C8 in TeX
Character table of C9⋊C8 in TeX

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