Copied to
clipboard

G = C12×D9order 216 = 23·33

Direct product of C12 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C12×D9
 Chief series C1 — C3 — C9 — C18 — C3×C18 — C6×D9 — C12×D9
 Lower central C9 — C12×D9
 Upper central C1 — C12

Generators and relations for C12×D9
G = < a,b,c | a12=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C12×D9
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 66 30 5 70 34 9 62 26)(2 67 31 6 71 35 10 63 27)(3 68 32 7 72 36 11 64 28)(4 69 33 8 61 25 12 65 29)(13 45 49 21 41 57 17 37 53)(14 46 50 22 42 58 18 38 54)(15 47 51 23 43 59 19 39 55)(16 48 52 24 44 60 20 40 56)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 36)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(23 34)(24 35)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 61)(47 62)(48 63)

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

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

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

C12×D9 is a maximal subgroup of   C36.39D6  Dic65D9  D125D9  D6.D18

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 9A ··· 9I 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 18A ··· 18I 36A ··· 36R order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 9 ··· 9 12 12 12 12 12 ··· 12 12 12 12 12 18 ··· 18 36 ··· 36 size 1 1 9 9 1 1 2 2 2 1 1 9 9 1 1 2 2 2 9 9 9 9 2 ··· 2 1 1 1 1 2 ··· 2 9 9 9 9 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D6 D9 C3×S3 C4×S3 D18 S3×C6 C3×D9 C4×D9 S3×C12 C6×D9 C12×D9 kernel C12×D9 C3×Dic9 C3×C36 C6×D9 C4×D9 C3×D9 Dic9 C36 D18 D9 C3×C12 C3×C6 C12 C12 C32 C6 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 8 1 1 3 2 2 3 2 6 6 4 6 12

Matrix representation of C12×D9 in GL2(𝔽37) generated by

 8 0 0 8
,
 5 6 6 0
,
 0 31 6 0
G:=sub<GL(2,GF(37))| [8,0,0,8],[5,6,6,0],[0,6,31,0] >;

C12×D9 in GAP, Magma, Sage, TeX

C_{12}\times D_9
% in TeX

G:=Group("C12xD9");
// GroupNames label

G:=SmallGroup(216,45);
// by ID

G=gap.SmallGroup(216,45);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,79,3604,208,5189]);
// Polycyclic

G:=Group<a,b,c|a^12=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

׿
×
𝔽