metabelian, supersoluble, monomial
Aliases: D6.7D18, C36.36D6, Dic18⋊5S3, C12.26D18, Dic9.4D6, Dic3.11D18, C12.50S32, (C4×S3)⋊2D9, (S3×C36)⋊4C2, C4.14(S3×D9), C36⋊S3⋊3C2, C9⋊D12⋊2C2, (S3×C12).6S3, (S3×C6).28D6, (C3×C12).97D6, C9⋊1(Q8⋊3S3), (C3×Dic18)⋊4C2, C18.D6⋊4C2, (C3×C36).7C22, C3⋊1(D36⋊5C2), C6.10(C22×D9), C18.10(C22×S3), (C3×C18).10C23, (C3×Dic3).38D6, (S3×C18).11C22, C3.2(D6.6D6), C32.4(C4○D12), (C3×Dic9).4C22, (C9×Dic3).12C22, C6.29(C2×S32), C2.14(C2×S3×D9), (C3×C9)⋊7(C4○D4), (C2×C9⋊S3).2C22, (C3×C6).78(C22×S3), SmallGroup(432,289)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic9.D6
G = < a,b,c,d | a18=1, b2=c6=d2=a9, bab-1=a-1, ac=ca, ad=da, cbc-1=a9b, bd=db, dcd-1=c5 >
Subgroups: 960 in 136 conjugacy classes, 41 normal (31 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×Q8, C3×C9, Dic9, C36, C36, D18, C2×C18, C3×Dic3, C3×Dic3, C3×C12, S3×C6, C2×C3⋊S3, C4○D12, Q8⋊3S3, S3×C9, C9⋊S3, C3×C18, Dic18, C4×D9, D36, C9⋊D4, C2×C36, C6.D6, C3⋊D12, C3×Dic6, S3×C12, C12⋊S3, C3×Dic9, C9×Dic3, C3×C36, S3×C18, C2×C9⋊S3, D36⋊5C2, D6.6D6, C18.D6, C9⋊D12, C3×Dic18, S3×C36, C36⋊S3, Dic9.D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, S32, C4○D12, Q8⋊3S3, C22×D9, C2×S32, S3×D9, D36⋊5C2, D6.6D6, C2×S3×D9, Dic9.D6
►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 51 10 42)(2 50 11 41)(3 49 12 40)(4 48 13 39)(5 47 14 38)(6 46 15 37)(7 45 16 54)(8 44 17 53)(9 43 18 52)(19 66 28 57)(20 65 29 56)(21 64 30 55)(22 63 31 72)(23 62 32 71)(24 61 33 70)(25 60 34 69)(26 59 35 68)(27 58 36 67)
(1 31 4 34 7 19 10 22 13 25 16 28)(2 32 5 35 8 20 11 23 14 26 17 29)(3 33 6 36 9 21 12 24 15 27 18 30)(37 67 52 64 49 61 46 58 43 55 40 70)(38 68 53 65 50 62 47 59 44 56 41 71)(39 69 54 66 51 63 48 60 45 57 42 72)
(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 69 28 60)(20 70 29 61)(21 71 30 62)(22 72 31 63)(23 55 32 64)(24 56 33 65)(25 57 34 66)(26 58 35 67)(27 59 36 68)
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), (1,51,10,42)(2,50,11,41)(3,49,12,40)(4,48,13,39)(5,47,14,38)(6,46,15,37)(7,45,16,54)(8,44,17,53)(9,43,18,52)(19,66,28,57)(20,65,29,56)(21,64,30,55)(22,63,31,72)(23,62,32,71)(24,61,33,70)(25,60,34,69)(26,59,35,68)(27,58,36,67), (1,31,4,34,7,19,10,22,13,25,16,28)(2,32,5,35,8,20,11,23,14,26,17,29)(3,33,6,36,9,21,12,24,15,27,18,30)(37,67,52,64,49,61,46,58,43,55,40,70)(38,68,53,65,50,62,47,59,44,56,41,71)(39,69,54,66,51,63,48,60,45,57,42,72), (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)(19,69,28,60)(20,70,29,61)(21,71,30,62)(22,72,31,63)(23,55,32,64)(24,56,33,65)(25,57,34,66)(26,58,35,67)(27,59,36,68)>;
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), (1,51,10,42)(2,50,11,41)(3,49,12,40)(4,48,13,39)(5,47,14,38)(6,46,15,37)(7,45,16,54)(8,44,17,53)(9,43,18,52)(19,66,28,57)(20,65,29,56)(21,64,30,55)(22,63,31,72)(23,62,32,71)(24,61,33,70)(25,60,34,69)(26,59,35,68)(27,58,36,67), (1,31,4,34,7,19,10,22,13,25,16,28)(2,32,5,35,8,20,11,23,14,26,17,29)(3,33,6,36,9,21,12,24,15,27,18,30)(37,67,52,64,49,61,46,58,43,55,40,70)(38,68,53,65,50,62,47,59,44,56,41,71)(39,69,54,66,51,63,48,60,45,57,42,72), (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)(19,69,28,60)(20,70,29,61)(21,71,30,62)(22,72,31,63)(23,55,32,64)(24,56,33,65)(25,57,34,66)(26,58,35,67)(27,59,36,68) );
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)], [(1,51,10,42),(2,50,11,41),(3,49,12,40),(4,48,13,39),(5,47,14,38),(6,46,15,37),(7,45,16,54),(8,44,17,53),(9,43,18,52),(19,66,28,57),(20,65,29,56),(21,64,30,55),(22,63,31,72),(23,62,32,71),(24,61,33,70),(25,60,34,69),(26,59,35,68),(27,58,36,67)], [(1,31,4,34,7,19,10,22,13,25,16,28),(2,32,5,35,8,20,11,23,14,26,17,29),(3,33,6,36,9,21,12,24,15,27,18,30),(37,67,52,64,49,61,46,58,43,55,40,70),(38,68,53,65,50,62,47,59,44,56,41,71),(39,69,54,66,51,63,48,60,45,57,42,72)], [(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),(19,69,28,60),(20,70,29,61),(21,71,30,62),(22,72,31,63),(23,55,32,64),(24,56,33,65),(25,57,34,66),(26,58,35,67),(27,59,36,68)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 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 | 9 | 9 | 9 | 9 | 9 | 9 | 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 | 54 | 54 | 2 | 2 | 4 | 2 | 3 | 3 | 18 | 18 | 2 | 2 | 4 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | D9 | D18 | D18 | D18 | C4○D12 | D36⋊5C2 | S32 | Q8⋊3S3 | C2×S32 | S3×D9 | D6.6D6 | C2×S3×D9 | Dic9.D6 |
| kernel | Dic9.D6 | C18.D6 | C9⋊D12 | C3×Dic18 | S3×C36 | C36⋊S3 | Dic18 | S3×C12 | Dic9 | C36 | C3×Dic3 | C3×C12 | S3×C6 | C3×C9 | C4×S3 | Dic3 | C12 | D6 | C32 | C3 | C12 | C9 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 12 | 1 | 1 | 1 | 3 | 2 | 3 | 6 |
Matrix representation of Dic9.D6 ►in GL6(𝔽37)
| 0 | 36 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 31 |
| 0 | 0 | 0 | 0 | 6 | 20 |
| 6 | 0 | 0 | 0 | 0 | 0 |
| 31 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 5 | 10 | 0 | 0 | 0 | 0 |
| 27 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 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))| [0,1,0,0,0,0,36,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,6,0,0,0,0,31,20],[6,31,0,0,0,0,0,31,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[5,27,0,0,0,0,10,32,0,0,0,0,0,0,0,36,0,0,0,0,1,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,36,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
Dic9.D6 in GAP, Magma, Sage, TeX
{\rm Dic}_9.D_6 % in TeX
G:=Group("Dic9.D6"); // GroupNames label
G:=SmallGroup(432,289);
// by ID
G=gap.SmallGroup(432,289);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,64,135,58,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^18=1,b^2=c^6=d^2=a^9,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^9*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations