metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊8D18, Q8⋊8D18, C9⋊22+ 1+4, D36⋊11C22, C36.26C23, C18.12C24, D18.7C23, Dic9.7C23, Dic18⋊12C22, C4○D4⋊5D9, (D4×D9)⋊5C2, (C2×C4)⋊4D18, C3.(D4○D12), (C2×D36)⋊13C2, (C2×C36)⋊5C22, Q8⋊3D9⋊5C2, (C3×D4).39D6, (C4×D9)⋊2C22, (D4×C9)⋊9C22, C9⋊D4⋊5C22, (C3×Q8).63D6, D36⋊5C2⋊8C2, (Q8×C9)⋊8C22, (C2×C12).105D6, (C2×C18).4C23, C6.49(S3×C23), C4.33(C22×D9), C2.13(C23×D9), (C22×D9)⋊4C22, C22.3(C22×D9), C12.187(C22×S3), (C9×C4○D4)⋊4C2, (C3×C4○D4).16S3, (C2×C6).10(C22×S3), SmallGroup(288,363)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊8D18
G = < a,b,c,d | a4=b2=c18=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, bd=db, dcd=c-1 >
Subgroups: 1200 in 249 conjugacy classes, 102 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C9, Dic3, C12, C12, D6, C2×C6, C2×D4, C4○D4, C4○D4, D9, C18, C18, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, 2+ 1+4, Dic9, C36, C36, D18, D18, C2×C18, C2×D12, C4○D12, S3×D4, Q8⋊3S3, C3×C4○D4, Dic18, C4×D9, D36, C9⋊D4, C2×C36, D4×C9, Q8×C9, C22×D9, D4○D12, C2×D36, D36⋊5C2, D4×D9, Q8⋊3D9, C9×C4○D4, D4⋊8D18
Quotients: C1, C2, C22, S3, C23, D6, C24, D9, C22×S3, 2+ 1+4, D18, S3×C23, C22×D9, D4○D12, C23×D9, D4⋊8D18
►On 72 points
(1 28 63 44)(2 29 64 45)(3 30 65 46)(4 31 66 47)(5 32 67 48)(6 33 68 49)(7 34 69 50)(8 35 70 51)(9 36 71 52)(10 19 72 53)(11 20 55 54)(12 21 56 37)(13 22 57 38)(14 23 58 39)(15 24 59 40)(16 25 60 41)(17 26 61 42)(18 27 62 43)
(2 64)(4 66)(6 68)(8 70)(10 72)(12 56)(14 58)(16 60)(18 62)(20 54)(22 38)(24 40)(26 42)(28 44)(30 46)(32 48)(34 50)(36 52)
(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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(37 69)(38 68)(39 67)(40 66)(41 65)(42 64)(43 63)(44 62)(45 61)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 72)(53 71)(54 70)
G:=sub<Sym(72)| (1,28,63,44)(2,29,64,45)(3,30,65,46)(4,31,66,47)(5,32,67,48)(6,33,68,49)(7,34,69,50)(8,35,70,51)(9,36,71,52)(10,19,72,53)(11,20,55,54)(12,21,56,37)(13,22,57,38)(14,23,58,39)(15,24,59,40)(16,25,60,41)(17,26,61,42)(18,27,62,43), (2,64)(4,66)(6,68)(8,70)(10,72)(12,56)(14,58)(16,60)(18,62)(20,54)(22,38)(24,40)(26,42)(28,44)(30,46)(32,48)(34,50)(36,52), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(37,69)(38,68)(39,67)(40,66)(41,65)(42,64)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,72)(53,71)(54,70)>;
G:=Group( (1,28,63,44)(2,29,64,45)(3,30,65,46)(4,31,66,47)(5,32,67,48)(6,33,68,49)(7,34,69,50)(8,35,70,51)(9,36,71,52)(10,19,72,53)(11,20,55,54)(12,21,56,37)(13,22,57,38)(14,23,58,39)(15,24,59,40)(16,25,60,41)(17,26,61,42)(18,27,62,43), (2,64)(4,66)(6,68)(8,70)(10,72)(12,56)(14,58)(16,60)(18,62)(20,54)(22,38)(24,40)(26,42)(28,44)(30,46)(32,48)(34,50)(36,52), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(37,69)(38,68)(39,67)(40,66)(41,65)(42,64)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,72)(53,71)(54,70) );
G=PermutationGroup([[(1,28,63,44),(2,29,64,45),(3,30,65,46),(4,31,66,47),(5,32,67,48),(6,33,68,49),(7,34,69,50),(8,35,70,51),(9,36,71,52),(10,19,72,53),(11,20,55,54),(12,21,56,37),(13,22,57,38),(14,23,58,39),(15,24,59,40),(16,25,60,41),(17,26,61,42),(18,27,62,43)], [(2,64),(4,66),(6,68),(8,70),(10,72),(12,56),(14,58),(16,60),(18,62),(20,54),(22,38),(24,40),(26,42),(28,44),(30,46),(32,48),(34,50),(36,52)], [(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)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,36),(11,35),(12,34),(13,33),(14,32),(15,31),(16,30),(17,29),(18,28),(37,69),(38,68),(39,67),(40,66),(41,65),(42,64),(43,63),(44,62),(45,61),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,72),(53,71),(54,70)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 18A | 18B | 18C | 18D | ··· | 18L | 36A | ··· | 36F | 36G | ··· | 36O |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 2 | 18 | ··· | 18 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D9 | D18 | D18 | D18 | 2+ 1+4 | D4○D12 | D4⋊8D18 |
| kernel | D4⋊8D18 | C2×D36 | D36⋊5C2 | D4×D9 | Q8⋊3D9 | C9×C4○D4 | C3×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C9 | C3 | C1 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 1 | 3 | 3 | 1 | 3 | 9 | 9 | 3 | 1 | 2 | 6 |
Matrix representation of D4⋊8D18 ►in GL4(𝔽37) generated by
| 36 | 0 | 5 | 14 |
| 0 | 36 | 23 | 28 |
| 6 | 34 | 1 | 0 |
| 3 | 9 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 31 | 3 | 36 | 0 |
| 34 | 28 | 0 | 36 |
| 25 | 33 | 4 | 21 |
| 4 | 29 | 16 | 20 |
| 10 | 0 | 12 | 4 |
| 0 | 10 | 33 | 8 |
| 8 | 33 | 0 | 0 |
| 25 | 29 | 0 | 0 |
| 27 | 0 | 25 | 33 |
| 10 | 10 | 8 | 12 |
G:=sub<GL(4,GF(37))| [36,0,6,3,0,36,34,9,5,23,1,0,14,28,0,1],[1,0,31,34,0,1,3,28,0,0,36,0,0,0,0,36],[25,4,10,0,33,29,0,10,4,16,12,33,21,20,4,8],[8,25,27,10,33,29,0,10,0,0,25,8,0,0,33,12] >;
D4⋊8D18 in GAP, Magma, Sage, TeX
D_4\rtimes_8D_{18} % in TeX
G:=Group("D4:8D18"); // GroupNames label
G:=SmallGroup(288,363);
// by ID
G=gap.SmallGroup(288,363);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^18=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations