metabelian, supersoluble, monomial
Aliases: Dic6⋊1D9, C36.14D6, C18.14D12, C12.32D18, C9⋊C8⋊3S3, C12.6S32, C4.3(S3×D9), (C3×C9)⋊6SD16, C9⋊2(C24⋊C2), (C9×Dic6)⋊2C2, (C3×C18).10D4, (C3×C12).78D6, C6.3(C9⋊D4), C36⋊S3.3C2, C3⋊1(Q8⋊2D9), (C3×Dic6).2S3, C2.6(C9⋊D12), (C3×C36).13C22, C6.16(C3⋊D12), C3.2(C32⋊5SD16), C32.3(Q8⋊2S3), (C3×C9⋊C8)⋊3C2, (C3×C6).46(C3⋊D4), SmallGroup(432,73)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C18.D12
G = < a,b,c | a12=c2=1, b18=a6, bab-1=cac=a-1, cbc=a3b17 >
Subgroups: 676 in 76 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, SD16, D9, C18, C18, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, D12, C3×Q8, C3×C9, C36, C36, D18, C3×Dic3, C3×C12, C2×C3⋊S3, C24⋊C2, Q8⋊2S3, C9⋊S3, C3×C18, C9⋊C8, D36, Q8×C9, C3×C3⋊C8, C3×Dic6, C12⋊S3, C9×Dic3, C3×C36, C2×C9⋊S3, Q8⋊2D9, C32⋊5SD16, C3×C9⋊C8, C9×Dic6, C36⋊S3, C18.D12
Quotients: C1, C2, C22, S3, D4, D6, SD16, D9, D12, C3⋊D4, D18, S32, C24⋊C2, Q8⋊2S3, C9⋊D4, C3⋊D12, S3×D9, Q8⋊2D9, C32⋊5SD16, C9⋊D12, C18.D12
►On 72 points
(1 40 7 46 13 52 19 58 25 64 31 70)(2 71 32 65 26 59 20 53 14 47 8 41)(3 42 9 48 15 54 21 60 27 66 33 72)(4 37 34 67 28 61 22 55 16 49 10 43)(5 44 11 50 17 56 23 62 29 68 35 38)(6 39 36 69 30 63 24 57 18 51 12 45)
(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 72)(2 26)(3 70)(4 24)(5 68)(6 22)(7 66)(8 20)(9 64)(10 18)(11 62)(12 16)(13 60)(15 58)(17 56)(19 54)(21 52)(23 50)(25 48)(27 46)(28 36)(29 44)(30 34)(31 42)(33 40)(35 38)(37 63)(39 61)(41 59)(43 57)(45 55)(47 53)(49 51)(65 71)(67 69)
G:=sub<Sym(72)| (1,40,7,46,13,52,19,58,25,64,31,70)(2,71,32,65,26,59,20,53,14,47,8,41)(3,42,9,48,15,54,21,60,27,66,33,72)(4,37,34,67,28,61,22,55,16,49,10,43)(5,44,11,50,17,56,23,62,29,68,35,38)(6,39,36,69,30,63,24,57,18,51,12,45), (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), (1,72)(2,26)(3,70)(4,24)(5,68)(6,22)(7,66)(8,20)(9,64)(10,18)(11,62)(12,16)(13,60)(15,58)(17,56)(19,54)(21,52)(23,50)(25,48)(27,46)(28,36)(29,44)(30,34)(31,42)(33,40)(35,38)(37,63)(39,61)(41,59)(43,57)(45,55)(47,53)(49,51)(65,71)(67,69)>;
G:=Group( (1,40,7,46,13,52,19,58,25,64,31,70)(2,71,32,65,26,59,20,53,14,47,8,41)(3,42,9,48,15,54,21,60,27,66,33,72)(4,37,34,67,28,61,22,55,16,49,10,43)(5,44,11,50,17,56,23,62,29,68,35,38)(6,39,36,69,30,63,24,57,18,51,12,45), (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), (1,72)(2,26)(3,70)(4,24)(5,68)(6,22)(7,66)(8,20)(9,64)(10,18)(11,62)(12,16)(13,60)(15,58)(17,56)(19,54)(21,52)(23,50)(25,48)(27,46)(28,36)(29,44)(30,34)(31,42)(33,40)(35,38)(37,63)(39,61)(41,59)(43,57)(45,55)(47,53)(49,51)(65,71)(67,69) );
G=PermutationGroup([[(1,40,7,46,13,52,19,58,25,64,31,70),(2,71,32,65,26,59,20,53,14,47,8,41),(3,42,9,48,15,54,21,60,27,66,33,72),(4,37,34,67,28,61,22,55,16,49,10,43),(5,44,11,50,17,56,23,62,29,68,35,38),(6,39,36,69,30,63,24,57,18,51,12,45)], [(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)], [(1,72),(2,26),(3,70),(4,24),(5,68),(6,22),(7,66),(8,20),(9,64),(10,18),(11,62),(12,16),(13,60),(15,58),(17,56),(19,54),(21,52),(23,50),(25,48),(27,46),(28,36),(29,44),(30,34),(31,42),(33,40),(35,38),(37,63),(39,61),(41,59),(43,57),(45,55),(47,53),(49,51),(65,71),(67,69)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 8A | 8B | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 18A | 18B | 18C | 18D | 18E | 18F | 24A | 24B | 24C | 24D | 36A | ··· | 36I | 36J | ··· | 36O |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 108 | 2 | 2 | 4 | 2 | 12 | 2 | 2 | 4 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 18 | 18 | 18 | 18 | 4 | ··· | 4 | 12 | ··· | 12 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | SD16 | D9 | D12 | C3⋊D4 | D18 | C24⋊C2 | C9⋊D4 | S32 | Q8⋊2S3 | C3⋊D12 | S3×D9 | Q8⋊2D9 | C32⋊5SD16 | C9⋊D12 | C18.D12 |
| kernel | C18.D12 | C3×C9⋊C8 | C9×Dic6 | C36⋊S3 | C9⋊C8 | C3×Dic6 | C3×C18 | C36 | C3×C12 | C3×C9 | Dic6 | C18 | C3×C6 | C12 | C9 | C6 | C12 | C32 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 2 | 3 | 4 | 6 | 1 | 1 | 1 | 3 | 3 | 2 | 3 | 6 |
Matrix representation of C18.D12 ►in GL6(𝔽73)
| 72 | 71 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 61 | 61 | 0 | 0 | 0 | 0 |
| 6 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 42 |
| 0 | 0 | 0 | 0 | 31 | 70 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 31 |
| 0 | 0 | 0 | 0 | 3 | 28 |
G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,71,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[61,6,0,0,0,0,61,12,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,28,31,0,0,0,0,42,70],[1,0,0,0,0,0,2,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,45,3,0,0,0,0,31,28] >;
C18.D12 in GAP, Magma, Sage, TeX
C_{18}.D_{12} % in TeX
G:=Group("C18.D12"); // GroupNames label
G:=SmallGroup(432,73);
// by ID
G=gap.SmallGroup(432,73);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,36,254,58,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c|a^12=c^2=1,b^18=a^6,b*a*b^-1=c*a*c=a^-1,c*b*c=a^3*b^17>;
// generators/relations