metabelian, supersoluble, monomial
Aliases: D6.5D18, D18.7D6, C36.43D6, C12.43D18, Dic9.8D6, Dic3.8D18, C12.65S32, (C4×S3)⋊4D9, (C4×D9)⋊4S3, (S3×C36)⋊2C2, (C12×D9)⋊7C2, C4.28(S3×D9), C9⋊D12⋊7C2, D6⋊D9⋊7C2, C9⋊2(C4○D12), C3⋊D36⋊7C2, (S3×C12).2S3, (S3×C6).26D6, C6.8(C22×D9), C9⋊Dic6⋊7C2, (C3×C12).161D6, C18.8(C22×S3), (C3×C18).8C23, C3⋊2(D36⋊5C2), (C6×D9).8C22, (S3×C18).9C22, (C3×C36).42C22, (C3×Dic3).29D6, C3.1(D6.D6), C32.2(C4○D12), C9⋊Dic3.10C22, (C9×Dic3).8C22, (C3×Dic9).10C22, (C4×C9⋊S3)⋊7C2, C6.27(C2×S32), C2.12(C2×S3×D9), (C3×C9)⋊5(C4○D4), (C2×C9⋊S3).8C22, (C3×C6).76(C22×S3), SmallGroup(432,287)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.D18
G = < a,b,c,d | a18=b2=1, c6=d2=a9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a9b, dcd-1=c5 >
Subgroups: 860 in 136 conjugacy classes, 41 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C9, C9, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×C9, Dic9, Dic9, C36, C36, D18, D18, C2×C18, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C4○D12, C3×D9, S3×C9, C9⋊S3, C3×C18, Dic18, C4×D9, C4×D9, D36, C9⋊D4, C2×C36, D6⋊S3, C3⋊D12, C32⋊2Q8, S3×C12, S3×C12, C4×C3⋊S3, C3×Dic9, C9×Dic3, C9⋊Dic3, C3×C36, C6×D9, S3×C18, C2×C9⋊S3, D36⋊5C2, D6.D6, C9⋊Dic6, C3⋊D36, D6⋊D9, C9⋊D12, C12×D9, S3×C36, C4×C9⋊S3, D6.D18
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, S32, C4○D12, C22×D9, C2×S32, S3×D9, D36⋊5C2, D6.D6, C2×S3×D9, D6.D18
►On 72 points
(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 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 72)(29 71)(30 70)(31 69)(32 68)(33 67)(34 66)(35 65)(36 64)
(1 32 16 29 13 26 10 23 7 20 4 35)(2 33 17 30 14 27 11 24 8 21 5 36)(3 34 18 31 15 28 12 25 9 22 6 19)(37 61 40 64 43 67 46 70 49 55 52 58)(38 62 41 65 44 68 47 71 50 56 53 59)(39 63 42 66 45 69 48 72 51 57 54 60)
(1 39 10 48)(2 40 11 49)(3 41 12 50)(4 42 13 51)(5 43 14 52)(6 44 15 53)(7 45 16 54)(8 46 17 37)(9 47 18 38)(19 56 28 65)(20 57 29 66)(21 58 30 67)(22 59 31 68)(23 60 32 69)(24 61 33 70)(25 62 34 71)(26 63 35 72)(27 64 36 55)
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), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,72)(29,71)(30,70)(31,69)(32,68)(33,67)(34,66)(35,65)(36,64), (1,32,16,29,13,26,10,23,7,20,4,35)(2,33,17,30,14,27,11,24,8,21,5,36)(3,34,18,31,15,28,12,25,9,22,6,19)(37,61,40,64,43,67,46,70,49,55,52,58)(38,62,41,65,44,68,47,71,50,56,53,59)(39,63,42,66,45,69,48,72,51,57,54,60), (1,39,10,48)(2,40,11,49)(3,41,12,50)(4,42,13,51)(5,43,14,52)(6,44,15,53)(7,45,16,54)(8,46,17,37)(9,47,18,38)(19,56,28,65)(20,57,29,66)(21,58,30,67)(22,59,31,68)(23,60,32,69)(24,61,33,70)(25,62,34,71)(26,63,35,72)(27,64,36,55)>;
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), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,72)(29,71)(30,70)(31,69)(32,68)(33,67)(34,66)(35,65)(36,64), (1,32,16,29,13,26,10,23,7,20,4,35)(2,33,17,30,14,27,11,24,8,21,5,36)(3,34,18,31,15,28,12,25,9,22,6,19)(37,61,40,64,43,67,46,70,49,55,52,58)(38,62,41,65,44,68,47,71,50,56,53,59)(39,63,42,66,45,69,48,72,51,57,54,60), (1,39,10,48)(2,40,11,49)(3,41,12,50)(4,42,13,51)(5,43,14,52)(6,44,15,53)(7,45,16,54)(8,46,17,37)(9,47,18,38)(19,56,28,65)(20,57,29,66)(21,58,30,67)(22,59,31,68)(23,60,32,69)(24,61,33,70)(25,62,34,71)(26,63,35,72)(27,64,36,55) );
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)], [(1,44),(2,43),(3,42),(4,41),(5,40),(6,39),(7,38),(8,37),(9,54),(10,53),(11,52),(12,51),(13,50),(14,49),(15,48),(16,47),(17,46),(18,45),(19,63),(20,62),(21,61),(22,60),(23,59),(24,58),(25,57),(26,56),(27,55),(28,72),(29,71),(30,70),(31,69),(32,68),(33,67),(34,66),(35,65),(36,64)], [(1,32,16,29,13,26,10,23,7,20,4,35),(2,33,17,30,14,27,11,24,8,21,5,36),(3,34,18,31,15,28,12,25,9,22,6,19),(37,61,40,64,43,67,46,70,49,55,52,58),(38,62,41,65,44,68,47,71,50,56,53,59),(39,63,42,66,45,69,48,72,51,57,54,60)], [(1,39,10,48),(2,40,11,49),(3,41,12,50),(4,42,13,51),(5,43,14,52),(6,44,15,53),(7,45,16,54),(8,46,17,37),(9,47,18,38),(19,56,28,65),(20,57,29,66),(21,58,30,67),(22,59,31,68),(23,60,32,69),(24,61,33,70),(25,62,34,71),(26,63,35,72),(27,64,36,55)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 18A | 18B | 18C | 18D | 18E | 18F | 18G | ··· | 18L | 36A | ··· | 36F | 36G | ··· | 36L | 36M | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 6 | 18 | 54 | 2 | 2 | 4 | 1 | 1 | 6 | 18 | 54 | 2 | 2 | 4 | 6 | 6 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | D6 | D6 | D6 | C4○D4 | D9 | D18 | D18 | D18 | C4○D12 | C4○D12 | D36⋊5C2 | S32 | C2×S32 | S3×D9 | D6.D6 | C2×S3×D9 | D6.D18 |
| kernel | D6.D18 | C9⋊Dic6 | C3⋊D36 | D6⋊D9 | C9⋊D12 | C12×D9 | S3×C36 | C4×C9⋊S3 | C4×D9 | S3×C12 | Dic9 | C36 | D18 | C3×Dic3 | C3×C12 | S3×C6 | C3×C9 | C4×S3 | Dic3 | C12 | D6 | C9 | C32 | C3 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 12 | 1 | 1 | 3 | 2 | 3 | 6 |
Matrix representation of D6.D18 ►in GL6(𝔽37)
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 11 |
| 0 | 0 | 0 | 0 | 26 | 6 |
| 20 | 2 | 0 | 0 | 0 | 0 |
| 4 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 26 |
| 0 | 0 | 0 | 0 | 6 | 17 |
| 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 20 | 2 | 0 | 0 | 0 | 0 |
| 3 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,26,0,0,0,0,11,6],[20,4,0,0,0,0,2,17,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,20,6,0,0,0,0,26,17],[6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,36,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[20,3,0,0,0,0,2,17,0,0,0,0,0,0,0,36,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D6.D18 in GAP, Magma, Sage, TeX
D_6.D_{18} % in TeX
G:=Group("D6.D18"); // GroupNames label
G:=SmallGroup(432,287);
// by ID
G=gap.SmallGroup(432,287);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,58,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^18=b^2=1,c^6=d^2=a^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^9*b,d*c*d^-1=c^5>;
// generators/relations