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

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

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

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

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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