Copied to
clipboard

## G = C8×D9order 144 = 24·32

### Direct product of C8 and D9

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

 Derived series C1 — C9 — C8×D9
 Chief series C1 — C3 — C9 — C18 — C36 — C4×D9 — C8×D9
 Lower central C9 — C8×D9
 Upper central C1 — C8

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

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 2A 2B 2C 3 4A 4B 4C 4D 6 8A 8B 8C 8D 8E 8F 8G 8H 9A 9B 9C 12A 12B 18A 18B 18C 24A 24B 24C 24D 36A ··· 36F 72A ··· 72L order 1 2 2 2 3 4 4 4 4 6 8 8 8 8 8 8 8 8 9 9 9 12 12 18 18 18 24 24 24 24 36 ··· 36 72 ··· 72 size 1 1 9 9 2 1 1 9 9 2 1 1 1 1 9 9 9 9 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 S3 D6 D9 C4×S3 D18 S3×C8 C4×D9 C8×D9 kernel C8×D9 C9⋊C8 C72 C4×D9 Dic9 D18 D9 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 2 8 1 1 3 2 3 4 6 12

Matrix representation of C8×D9 in GL2(𝔽17) generated by

 8 0 0 8
,
 14 8 2 0
,
 0 9 2 0
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

׿
×
𝔽