metabelian, supersoluble, monomial
Aliases: Dic6⋊5D9, D18.5D6, C36.23D6, C12.35D18, Dic9.10D6, Dic3.3D18, C12.15S32, (C4×D9)⋊1S3, C4.6(S3×D9), (C12×D9)⋊1C2, C36⋊S3⋊7C2, C9⋊1(C4○D12), C3⋊D36⋊4C2, (C9×Dic6)⋊5C2, (C3×C12).91D6, C6.3(C22×D9), C3⋊1(Q8⋊3D9), C18.D6⋊1C2, C18.3(C22×S3), (C3×C18).3C23, (C3×Dic6).7S3, (C3×Dic3).3D6, (C6×D9).6C22, (C3×C36).26C22, C3.1(D6.6D6), (C3×Dic9).8C22, (C9×Dic3).3C22, C32.3(Q8⋊3S3), C2.7(C2×S3×D9), C6.22(C2×S32), (C3×C9)⋊2(C4○D4), (C2×C9⋊S3).1C22, (C3×C6).71(C22×S3), SmallGroup(432,282)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊5D9
G = < a,b,c,d | a12=c9=d2=1, b2=a6, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a6b, dcd=c-1 >
Subgroups: 988 in 136 conjugacy classes, 41 normal (29 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C9, C9, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, C3×C9, Dic9, C36, C36, D18, D18, C3×Dic3, C3×Dic3, C3×C12, S3×C6, C2×C3⋊S3, C4○D12, Q8⋊3S3, C3×D9, C9⋊S3, C3×C18, C4×D9, C4×D9, D36, Q8×C9, C6.D6, C3⋊D12, C3×Dic6, S3×C12, C12⋊S3, C3×Dic9, C9×Dic3, C3×C36, C6×D9, C2×C9⋊S3, Q8⋊3D9, D6.6D6, C18.D6, C3⋊D36, C9×Dic6, C12×D9, C36⋊S3, Dic6⋊5D9
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, S32, C4○D12, Q8⋊3S3, C22×D9, C2×S32, S3×D9, Q8⋊3D9, D6.6D6, C2×S3×D9, Dic6⋊5D9
►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 16 7 22)(2 15 8 21)(3 14 9 20)(4 13 10 19)(5 24 11 18)(6 23 12 17)(25 60 31 54)(26 59 32 53)(27 58 33 52)(28 57 34 51)(29 56 35 50)(30 55 36 49)(37 64 43 70)(38 63 44 69)(39 62 45 68)(40 61 46 67)(41 72 47 66)(42 71 48 65)
(1 72 25 5 64 29 9 68 33)(2 61 26 6 65 30 10 69 34)(3 62 27 7 66 31 11 70 35)(4 63 28 8 67 32 12 71 36)(13 44 57 21 40 53 17 48 49)(14 45 58 22 41 54 18 37 50)(15 46 59 23 42 55 19 38 51)(16 47 60 24 43 56 20 39 52)
(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 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 25)(24 26)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)
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), (1,16,7,22)(2,15,8,21)(3,14,9,20)(4,13,10,19)(5,24,11,18)(6,23,12,17)(25,60,31,54)(26,59,32,53)(27,58,33,52)(28,57,34,51)(29,56,35,50)(30,55,36,49)(37,64,43,70)(38,63,44,69)(39,62,45,68)(40,61,46,67)(41,72,47,66)(42,71,48,65), (1,72,25,5,64,29,9,68,33)(2,61,26,6,65,30,10,69,34)(3,62,27,7,66,31,11,70,35)(4,63,28,8,67,32,12,71,36)(13,44,57,21,40,53,17,48,49)(14,45,58,22,41,54,18,37,50)(15,46,59,23,42,55,19,38,51)(16,47,60,24,43,56,20,39,52), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,25)(24,26)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)>;
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), (1,16,7,22)(2,15,8,21)(3,14,9,20)(4,13,10,19)(5,24,11,18)(6,23,12,17)(25,60,31,54)(26,59,32,53)(27,58,33,52)(28,57,34,51)(29,56,35,50)(30,55,36,49)(37,64,43,70)(38,63,44,69)(39,62,45,68)(40,61,46,67)(41,72,47,66)(42,71,48,65), (1,72,25,5,64,29,9,68,33)(2,61,26,6,65,30,10,69,34)(3,62,27,7,66,31,11,70,35)(4,63,28,8,67,32,12,71,36)(13,44,57,21,40,53,17,48,49)(14,45,58,22,41,54,18,37,50)(15,46,59,23,42,55,19,38,51)(16,47,60,24,43,56,20,39,52), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,25)(24,26)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66) );
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)], [(1,16,7,22),(2,15,8,21),(3,14,9,20),(4,13,10,19),(5,24,11,18),(6,23,12,17),(25,60,31,54),(26,59,32,53),(27,58,33,52),(28,57,34,51),(29,56,35,50),(30,55,36,49),(37,64,43,70),(38,63,44,69),(39,62,45,68),(40,61,46,67),(41,72,47,66),(42,71,48,65)], [(1,72,25,5,64,29,9,68,33),(2,61,26,6,65,30,10,69,34),(3,62,27,7,66,31,11,70,35),(4,63,28,8,67,32,12,71,36),(13,44,57,21,40,53,17,48,49),(14,45,58,22,41,54,18,37,50),(15,46,59,23,42,55,19,38,51),(16,47,60,24,43,56,20,39,52)], [(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,27),(14,28),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,25),(24,26),(37,67),(38,68),(39,69),(40,70),(41,71),(42,72),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66)]])
54 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 | 36A | ··· | 36I | 36J | ··· | 36O |
| 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 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 18 | 54 | 54 | 2 | 2 | 4 | 2 | 6 | 6 | 9 | 9 | 2 | 2 | 4 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 |
54 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | D9 | D18 | D18 | C4○D12 | S32 | Q8⋊3S3 | C2×S32 | S3×D9 | Q8⋊3D9 | D6.6D6 | C2×S3×D9 | Dic6⋊5D9 |
| kernel | Dic6⋊5D9 | C18.D6 | C3⋊D36 | C9×Dic6 | C12×D9 | C36⋊S3 | C4×D9 | C3×Dic6 | Dic9 | C36 | D18 | C3×Dic3 | C3×C12 | C3×C9 | Dic6 | Dic3 | C12 | C9 | C12 | C32 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 3 | 6 | 3 | 4 | 1 | 1 | 1 | 3 | 3 | 2 | 3 | 6 |
Matrix representation of Dic6⋊5D9 ►in GL6(𝔽37)
| 1 | 35 | 0 | 0 | 0 | 0 |
| 1 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 31 | 0 | 0 | 0 | 0 | 0 |
| 31 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 24 |
| 0 | 0 | 0 | 0 | 15 | 6 |
| 6 | 25 | 0 | 0 | 0 | 0 |
| 6 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 13 |
| 0 | 0 | 0 | 0 | 12 | 17 |
G:=sub<GL(6,GF(37))| [1,1,0,0,0,0,35,36,0,0,0,0,0,0,1,36,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[31,31,0,0,0,0,0,6,0,0,0,0,0,0,1,36,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,15,0,0,0,0,24,6],[6,6,0,0,0,0,25,31,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,20,12,0,0,0,0,13,17] >;
Dic6⋊5D9 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_5D_9 % in TeX
G:=Group("Dic6:5D9"); // GroupNames label
G:=SmallGroup(432,282);
// by ID
G=gap.SmallGroup(432,282);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,254,58,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^9=d^2=1,b^2=a^6,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations