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G = C9×D8order 144 = 24·32

Direct product of C9 and D8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C9×D8, D4⋊C18, C81C18, C725C2, C24.2C6, C18.14D4, C36.17C22, C3.(C3×D8), (C3×D8).C3, (D4×C9)⋊4C2, C2.3(D4×C9), C4.1(C2×C18), (C3×D4).2C6, C6.14(C3×D4), C12.17(C2×C6), SmallGroup(144,25)

Series: Derived Chief Lower central Upper central

C1C4 — C9×D8
C1C2C6C12C36D4×C9 — C9×D8
C1C2C4 — C9×D8
C1C18C36 — C9×D8

Generators and relations for C9×D8
 G = < a,b,c | a9=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

4C2
4C2
2C22
2C22
4C6
4C6
2C2×C6
2C2×C6
4C18
4C18
2C2×C18
2C2×C18

Smallest permutation representation of C9×D8
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 28 53 23 42 66 59 11)(2 29 54 24 43 67 60 12)(3 30 46 25 44 68 61 13)(4 31 47 26 45 69 62 14)(5 32 48 27 37 70 63 15)(6 33 49 19 38 71 55 16)(7 34 50 20 39 72 56 17)(8 35 51 21 40 64 57 18)(9 36 52 22 41 65 58 10)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 10)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)(26 45)(27 37)(28 59)(29 60)(30 61)(31 62)(32 63)(33 55)(34 56)(35 57)(36 58)(46 68)(47 69)(48 70)(49 71)(50 72)(51 64)(52 65)(53 66)(54 67)

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

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

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

C9×D8 is a maximal subgroup of   C9⋊D16  D8.D9  D8⋊D9  D83D9

63 conjugacy classes

class 1 2A2B2C3A3B 4 6A6B6C6D6E6F8A8B9A···9F12A12B18A···18F18G···18R24A24B24C24D36A···36F72A···72L
order1222334666666889···9121218···1818···182424242436···3672···72
size1144112114444221···1221···14···422222···22···2

63 irreducible representations

dim111111111222222
type+++++
imageC1C2C2C3C6C6C9C18C18D4D8C3×D4C3×D8D4×C9C9×D8
kernelC9×D8C72D4×C9C3×D8C24C3×D4D8C8D4C18C9C6C3C2C1
# reps11222466121224612

Matrix representation of C9×D8 in GL2(𝔽73) generated by

320
032
,
1657
1616
,
1657
5757
G:=sub<GL(2,GF(73))| [32,0,0,32],[16,16,57,16],[16,57,57,57] >;

C9×D8 in GAP, Magma, Sage, TeX

C_9\times D_8
% in TeX

G:=Group("C9xD8");
// GroupNames label

G:=SmallGroup(144,25);
// by ID

G=gap.SmallGroup(144,25);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-2,169,122,2019,1017,165]);
// Polycyclic

G:=Group<a,b,c|a^9=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C9×D8 in TeX

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