direct product, metacyclic, supersoluble, monomial
Aliases: C9xD12, C36:3S3, C12:1C18, D6:1C18, C18.21D6, C4:(S3xC9), (C3xC9):4D4, C3:1(D4xC9), (C3xC36):6C2, (C3xD12).C3, (S3xC18):1C2, (S3xC6).1C6, C2.4(S3xC18), C6.31(S3xC6), C6.3(C2xC18), C3.4(C3xD12), C12.16(C3xS3), (C3xC12).11C6, C32.2(C3xD4), (C3xC18).10C22, (C3xC6).20(C2xC6), SmallGroup(216,48)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9xD12
G = < a,b,c | a9=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C3, C22, S3, C6, D4, C9, D6, C2xC6, C18, C3xS3, D12, C3xD4, C2xC18, S3xC6, S3xC9, D4xC9, C3xD12, S3xC18, C9xD12
Smallest permutation representation of C9xD12
►On 72 points
Generators in S72
(1 55 62 5 59 66 9 51 70)(2 56 63 6 60 67 10 52 71)(3 57 64 7 49 68 11 53 72)(4 58 65 8 50 69 12 54 61)(13 35 40 21 31 48 17 27 44)(14 36 41 22 32 37 18 28 45)(15 25 42 23 33 38 19 29 46)(16 26 43 24 34 39 20 30 47)
(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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 72)(22 71)(23 70)(24 69)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 60)(46 59)(47 58)(48 57)
G:=sub<Sym(72)| (1,55,62,5,59,66,9,51,70)(2,56,63,6,60,67,10,52,71)(3,57,64,7,49,68,11,53,72)(4,58,65,8,50,69,12,54,61)(13,35,40,21,31,48,17,27,44)(14,36,41,22,32,37,18,28,45)(15,25,42,23,33,38,19,29,46)(16,26,43,24,34,39,20,30,47), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,36)(11,35)(12,34)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,72)(22,71)(23,70)(24,69)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,60)(46,59)(47,58)(48,57)>;
G:=Group( (1,55,62,5,59,66,9,51,70)(2,56,63,6,60,67,10,52,71)(3,57,64,7,49,68,11,53,72)(4,58,65,8,50,69,12,54,61)(13,35,40,21,31,48,17,27,44)(14,36,41,22,32,37,18,28,45)(15,25,42,23,33,38,19,29,46)(16,26,43,24,34,39,20,30,47), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,36)(11,35)(12,34)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,72)(22,71)(23,70)(24,69)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,60)(46,59)(47,58)(48,57) );
G=PermutationGroup([[(1,55,62,5,59,66,9,51,70),(2,56,63,6,60,67,10,52,71),(3,57,64,7,49,68,11,53,72),(4,58,65,8,50,69,12,54,61),(13,35,40,21,31,48,17,27,44),(14,36,41,22,32,37,18,28,45),(15,25,42,23,33,38,19,29,46),(16,26,43,24,34,39,20,30,47)], [(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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,36),(11,35),(12,34),(13,68),(14,67),(15,66),(16,65),(17,64),(18,63),(19,62),(20,61),(21,72),(22,71),(23,70),(24,69),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,60),(46,59),(47,58),(48,57)]])
C9xD12 is a maximal subgroup of
D36:S3 C9:D24 D12.D9 C36.D6 D12:5D9 D12:D9 C36:D6 S3xD4xC9
81 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 9A | ··· | 9F | 9G | ··· | 9L | 12A | ··· | 12H | 18A | ··· | 18F | 18G | ··· | 18L | 18M | ··· | 18X | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
81 irreducible representations
| dim | 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 | C3 | C6 | C6 | C9 | C18 | C18 | S3 | D4 | D6 | C3xS3 | D12 | C3xD4 | S3xC6 | S3xC9 | D4xC9 | C3xD12 | S3xC18 | C9xD12 |
| kernel | C9xD12 | C3xC36 | S3xC18 | C3xD12 | C3xC12 | S3xC6 | D12 | C12 | D6 | C36 | C3xC9 | C18 | C12 | C9 | C32 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 4 | 6 | 12 |
Matrix representation of C9xD12 ►in GL2(F37) generated by
| 7 | 0 |
| 0 | 7 |
| 23 | 0 |
| 0 | 29 |
| 0 | 30 |
| 21 | 0 |
G:=sub<GL(2,GF(37))| [7,0,0,7],[23,0,0,29],[0,21,30,0] >;
C9xD12 in GAP, Magma, Sage, TeX
C_9\times D_{12} % in TeX
G:=Group("C9xD12"); // GroupNames label
G:=SmallGroup(216,48);
// by ID
G=gap.SmallGroup(216,48);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,79,122,5189]);
// Polycyclic
G:=Group<a,b,c|a^9=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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