direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C12×D9, C36⋊6C6, Dic9⋊5C6, D18.2C6, C6.20D18, (C3×C36)⋊4C2, C9⋊4(C2×C12), C6.7(S3×C6), C2.1(C6×D9), C3.1(S3×C12), (C6×D9).2C2, (C3×C6).46D6, C12.11(C3×S3), C18.10(C2×C6), (C3×C12).17S3, (C3×Dic9)⋊5C2, C32.4(C4×S3), (C3×C18).14C22, (C3×C9)⋊5(C2×C4), SmallGroup(216,45)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C12×D9 |
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 Dic6⋊5D9 D12⋊5D9 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
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