metabelian, supersoluble, monomial
Aliases: D18.4D6, D6.4D18, Dic9.6D6, C62.67D6, Dic3.6D18, C9⋊D4⋊1S3, C3⋊D4⋊1D9, (S3×C6).6D6, (C2×C18).5D6, (C2×C6).5D18, D6⋊D9⋊4C2, (Dic3×D9)⋊2C2, (S3×Dic9)⋊4C2, C9⋊4(D4⋊2S3), C3⋊4(D4⋊2D9), C22.3(S3×D9), C9⋊Dic6⋊6C2, (C3×Dic3).6D6, (C6×D9).4C22, C6.22(C22×D9), (S3×C18).4C22, (C3×C18).22C23, (C6×C18).16C22, C18.22(C22×S3), C3.1(D6.4D6), C9⋊Dic3.12C22, (C9×Dic3).6C22, (C3×Dic9).6C22, C32.5(D4⋊2S3), (C2×C6).5S32, C6.41(C2×S32), C2.22(C2×S3×D9), (C9×C3⋊D4)⋊2C2, (C3×C9⋊D4)⋊2C2, (C3×C9)⋊10(C4○D4), (C2×C9⋊Dic3)⋊7C2, (C3×C3⋊D4).2S3, (C3×C6).90(C22×S3), SmallGroup(432,310)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D18.4D6
G = < a,b,c,d | a18=b2=1, c6=d2=a9, bab=cac-1=dad-1=a-1, cbc-1=a7b, dbd-1=a16b, dcd-1=c5 >
Subgroups: 728 in 136 conjugacy classes, 41 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, Q8, C9, C9, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3⋊D4, C3×D4, C3×C9, Dic9, Dic9, C36, D18, C2×C18, C2×C18, C3×Dic3, C3×Dic3, C3⋊Dic3, S3×C6, S3×C6, C62, D4⋊2S3, C3×D9, S3×C9, C3×C18, C3×C18, Dic18, C4×D9, C2×Dic9, C9⋊D4, C9⋊D4, D4×C9, S3×Dic3, D6⋊S3, C32⋊2Q8, C3×C3⋊D4, C3×C3⋊D4, C2×C3⋊Dic3, C3×Dic9, C9×Dic3, C9⋊Dic3, C6×D9, S3×C18, C6×C18, D4⋊2D9, D6.4D6, C9⋊Dic6, Dic3×D9, S3×Dic9, D6⋊D9, C3×C9⋊D4, C9×C3⋊D4, C2×C9⋊Dic3, D18.4D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, S32, D4⋊2S3, C22×D9, C2×S32, S3×D9, D4⋊2D9, D6.4D6, C2×S3×D9, D18.4D6
►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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(37 56)(38 55)(39 72)(40 71)(41 70)(42 69)(43 68)(44 67)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)
(1 34 16 19 13 22 10 25 7 28 4 31)(2 33 17 36 14 21 11 24 8 27 5 30)(3 32 18 35 15 20 12 23 9 26 6 29)(37 69 40 66 43 63 46 60 49 57 52 72)(38 68 41 65 44 62 47 59 50 56 53 71)(39 67 42 64 45 61 48 58 51 55 54 70)
(1 63 10 72)(2 62 11 71)(3 61 12 70)(4 60 13 69)(5 59 14 68)(6 58 15 67)(7 57 16 66)(8 56 17 65)(9 55 18 64)(19 49 28 40)(20 48 29 39)(21 47 30 38)(22 46 31 37)(23 45 32 54)(24 44 33 53)(25 43 34 52)(26 42 35 51)(27 41 36 50)
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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(37,56)(38,55)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57), (1,34,16,19,13,22,10,25,7,28,4,31)(2,33,17,36,14,21,11,24,8,27,5,30)(3,32,18,35,15,20,12,23,9,26,6,29)(37,69,40,66,43,63,46,60,49,57,52,72)(38,68,41,65,44,62,47,59,50,56,53,71)(39,67,42,64,45,61,48,58,51,55,54,70), (1,63,10,72)(2,62,11,71)(3,61,12,70)(4,60,13,69)(5,59,14,68)(6,58,15,67)(7,57,16,66)(8,56,17,65)(9,55,18,64)(19,49,28,40)(20,48,29,39)(21,47,30,38)(22,46,31,37)(23,45,32,54)(24,44,33,53)(25,43,34,52)(26,42,35,51)(27,41,36,50)>;
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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(37,56)(38,55)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57), (1,34,16,19,13,22,10,25,7,28,4,31)(2,33,17,36,14,21,11,24,8,27,5,30)(3,32,18,35,15,20,12,23,9,26,6,29)(37,69,40,66,43,63,46,60,49,57,52,72)(38,68,41,65,44,62,47,59,50,56,53,71)(39,67,42,64,45,61,48,58,51,55,54,70), (1,63,10,72)(2,62,11,71)(3,61,12,70)(4,60,13,69)(5,59,14,68)(6,58,15,67)(7,57,16,66)(8,56,17,65)(9,55,18,64)(19,49,28,40)(20,48,29,39)(21,47,30,38)(22,46,31,37)(23,45,32,54)(24,44,33,53)(25,43,34,52)(26,42,35,51)(27,41,36,50) );
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)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(37,56),(38,55),(39,72),(40,71),(41,70),(42,69),(43,68),(44,67),(45,66),(46,65),(47,64),(48,63),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57)], [(1,34,16,19,13,22,10,25,7,28,4,31),(2,33,17,36,14,21,11,24,8,27,5,30),(3,32,18,35,15,20,12,23,9,26,6,29),(37,69,40,66,43,63,46,60,49,57,52,72),(38,68,41,65,44,62,47,59,50,56,53,71),(39,67,42,64,45,61,48,58,51,55,54,70)], [(1,63,10,72),(2,62,11,71),(3,61,12,70),(4,60,13,69),(5,59,14,68),(6,58,15,67),(7,57,16,66),(8,56,17,65),(9,55,18,64),(19,49,28,40),(20,48,29,39),(21,47,30,38),(22,46,31,37),(23,45,32,54),(24,44,33,53),(25,43,34,52),(26,42,35,51),(27,41,36,50)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6G | 6H | 6I | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 18A | 18B | 18C | 18D | ··· | 18O | 18P | 18Q | 18R | 36A | 36B | 36C |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 18 | 18 | 18 | 36 | 36 | 36 |
| size | 1 | 1 | 2 | 6 | 18 | 2 | 2 | 4 | 6 | 18 | 27 | 27 | 54 | 2 | 2 | 4 | ··· | 4 | 12 | 36 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | 36 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 12 | 12 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | S32 | D4⋊2S3 | D4⋊2S3 | C2×S32 | S3×D9 | D4⋊2D9 | D6.4D6 | C2×S3×D9 | D18.4D6 |
| kernel | D18.4D6 | C9⋊Dic6 | Dic3×D9 | S3×Dic9 | D6⋊D9 | C3×C9⋊D4 | C9×C3⋊D4 | C2×C9⋊Dic3 | C9⋊D4 | C3×C3⋊D4 | Dic9 | D18 | C2×C18 | C3×Dic3 | S3×C6 | C62 | C3×C9 | C3⋊D4 | Dic3 | D6 | C2×C6 | C2×C6 | C9 | C32 | C6 | C22 | C3 | 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 | 1 | 1 | 1 | 1 | 3 | 3 | 2 | 3 | 6 |
Matrix representation of D18.4D6 ►in GL6(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 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 | 0 | 7 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 1 | 36 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 0 | 0 | 31 |
G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,7,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,7,0,0,0,0,0,0,0,0,36,0,0,0,0,36,0],[1,1,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,1,0],[1,1,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,31,0,0,0,0,0,0,31] >;
D18.4D6 in GAP, Magma, Sage, TeX
D_{18}._4D_6 % in TeX
G:=Group("D18.4D6"); // GroupNames label
G:=SmallGroup(432,310);
// by ID
G=gap.SmallGroup(432,310);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^18=b^2=1,c^6=d^2=a^9,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^7*b,d*b*d^-1=a^16*b,d*c*d^-1=c^5>;
// generators/relations