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G = C8xD9order 144 = 24·32

Direct product of C8 and D9

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C8xD9, C72:3C2, C24.7S3, D18.2C4, C4.12D18, C12.50D6, Dic9.2C4, C36.12C22, C9:C8:6C2, C9:1(C2xC8), C3.(S3xC8), C6.5(C4xS3), C2.1(C4xD9), C18.1(C2xC4), (C4xD9).3C2, SmallGroup(144,5)

Series: Derived Chief Lower central Upper central

C1C9 — C8xD9
C1C3C9C18C36C4xD9 — C8xD9
C9 — C8xD9
C1C8

Generators and relations for C8xD9
 G = < a,b,c | a8=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 103 in 33 conjugacy classes, 19 normal (17 characteristic)
Quotients: C1, C2, C4, C22, S3, C8, C2xC4, D6, C2xC8, D9, C4xS3, D18, S3xC8, C4xD9, C8xD9
9C2
9C2
9C22
9C4
3S3
3S3
9C2xC4
9C8
3Dic3
3D6
9C2xC8
3C3:C8
3C4xS3
3S3xC8

Smallest permutation representation of C8xD9
On 72 points
Generators in S72
(1 68 32 50 14 59 23 41)(2 69 33 51 15 60 24 42)(3 70 34 52 16 61 25 43)(4 71 35 53 17 62 26 44)(5 72 36 54 18 63 27 45)(6 64 28 46 10 55 19 37)(7 65 29 47 11 56 20 38)(8 66 30 48 12 57 21 39)(9 67 31 49 13 58 22 40)
(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 13)(2 12)(3 11)(4 10)(5 18)(6 17)(7 16)(8 15)(9 14)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(27 36)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(45 54)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(63 72)

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

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

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

C8xD9 is a maximal subgroup of   C16:D9  D36.2C4  D36.C4  D8:3D9  SD16:3D9  D72:5C2  C36.38D6
C8xD9 is a maximal quotient of   C16:D9  Dic9:C8  D18:C8  C36.38D6

48 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 8A8B8C8D8E8F8G8H9A9B9C12A12B18A18B18C24A24B24C24D36A···36F72A···72L
order12223444468888888899912121818182424242436···3672···72
size1199211992111199992222222222222···22···2

48 irreducible representations

dim111111122222222
type++++++++
imageC1C2C2C2C4C4C8S3D6D9C4xS3D18S3xC8C4xD9C8xD9
kernelC8xD9C9:C8C72C4xD9Dic9D18D9C24C12C8C6C4C3C2C1
# reps1111228113234612

Matrix representation of C8xD9 in GL2(F17) generated by

80
08
,
148
20
,
09
20
G:=sub<GL(2,GF(17))| [8,0,0,8],[14,2,8,0],[0,2,9,0] >;

C8xD9 in GAP, Magma, Sage, TeX

C_8\times D_9
% in TeX

G:=Group("C8xD9");
// GroupNames label

G:=SmallGroup(144,5);
// by ID

G=gap.SmallGroup(144,5);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,31,50,2404,208,3461]);
// Polycyclic

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

Export

Subgroup lattice of C8xD9 in TeX

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