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G = C8⋊D9order 144 = 24·32

3rd semidirect product of C8 and D9 acting via D9/C9=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C83D9, C724C2, D18.C4, C24.8S3, Dic9.C4, C91M4(2), C4.13D18, C12.51D6, C36.13C22, C9⋊C84C2, C6.6(C4×S3), C2.3(C4×D9), C3.(C8⋊S3), C18.2(C2×C4), (C4×D9).2C2, SmallGroup(144,6)

Series: Derived Chief Lower central Upper central

C1C18 — C8⋊D9
C1C3C9C18C36C4×D9 — C8⋊D9
C9C18 — C8⋊D9
C1C4C8

Generators and relations for C8⋊D9
 G = < a,b,c | a8=b9=c2=1, ab=ba, cac=a5, cbc=b-1 >

18C2
9C4
9C22
6S3
9C8
9C2×C4
3D6
3Dic3
2D9
9M4(2)
3C4×S3
3C3⋊C8
3C8⋊S3

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 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)(28 35)(29 34)(30 33)(31 32)(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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(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)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,26),(20,25),(21,24),(22,23),(28,35),(29,34),(30,33),(31,32),(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
D36.2C4  M4(2)×D9  D36.C4  D8⋊D9  D72⋊C2  SD16⋊D9  Q16⋊D9  C8⋊D27  C36.39D6  C36.40D6  C72⋊C6  C72⋊S3
C8⋊D9 is a maximal quotient of
Dic9⋊C8  C72⋊C4  D18⋊C8  C8⋊D27  C36.39D6  C36.40D6  C72⋊S3

42 conjugacy classes

class 1 2A2B 3 4A4B4C 6 8A8B8C8D9A9B9C12A12B18A18B18C24A24B24C24D36A···36F72A···72L
order12234446888899912121818182424242436···3672···72
size11182111822218182222222222222···22···2

42 irreducible representations

dim111111222222222
type++++++++
imageC1C2C2C2C4C4S3D6M4(2)D9C4×S3D18C8⋊S3C4×D9C8⋊D9
kernelC8⋊D9C9⋊C8C72C4×D9Dic9D18C24C12C9C8C6C4C3C2C1
# reps1111221123234612

Matrix representation of C8⋊D9 in GL2(𝔽73) generated by

816
5765
,
2870
331
,
2870
4245
G:=sub<GL(2,GF(73))| [8,57,16,65],[28,3,70,31],[28,42,70,45] >;

C8⋊D9 in GAP, Magma, Sage, TeX

C_8\rtimes D_9
% in TeX

G:=Group("C8:D9");
// GroupNames label

G:=SmallGroup(144,6);
// by ID

G=gap.SmallGroup(144,6);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,31,50,2404,208,3461]);
// Polycyclic

G:=Group<a,b,c|a^8=b^9=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C8⋊D9 in TeX

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