Copied to
clipboard

G = C8×D9order 144 = 24·32

Direct product of C8 and D9

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

Aliases: C8×D9, C723C2, C24.7S3, D18.2C4, C4.12D18, C12.50D6, Dic9.2C4, C36.12C22, C9⋊C86C2, C91(C2×C8), C3.(S3×C8), C6.5(C4×S3), C2.1(C4×D9), C18.1(C2×C4), (C4×D9).3C2, SmallGroup(144,5)

Series: Derived Chief Lower central Upper central

C1C9 — C8×D9
C1C3C9C18C36C4×D9 — C8×D9
C9 — C8×D9
C1C8

Generators and relations for C8×D9
 G = < a,b,c | a8=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

9C2
9C2
9C22
9C4
3S3
3S3
9C2×C4
9C8
3Dic3
3D6
9C2×C8
3C3⋊C8
3C4×S3
3S3×C8

Smallest permutation representation of C8×D9
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)])

C8×D9 is a maximal subgroup of   C16⋊D9  D36.2C4  D36.C4  D83D9  SD163D9  D725C2  C36.38D6
C8×D9 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++++++++
imageC1C2C2C2C4C4C8S3D6D9C4×S3D18S3×C8C4×D9C8×D9
kernelC8×D9C9⋊C8C72C4×D9Dic9D18D9C24C12C8C6C4C3C2C1
# reps1111228113234612

Matrix representation of C8×D9 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] >;

C8×D9 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 C8×D9 in TeX

׿
×
𝔽