metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊4D18, Q8⋊5D18, C36.51D4, C36.18C23, D36.11C22, D4⋊D9⋊6C2, C4○D4⋊3D9, C9⋊C8⋊4C22, C9⋊5(C8⋊C22), (C2×C18).9D4, (C2×D36)⋊10C2, C3.(D4⋊D6), Q8⋊2D9⋊6C2, (C3×D4).33D6, C18.60(C2×D4), (C2×C12).69D6, (C2×C4).21D18, (D4×C9)⋊4C22, (C3×Q8).57D6, (Q8×C9)⋊4C22, C4.25(C9⋊D4), C4.Dic9⋊10C2, C4.18(C22×D9), (C2×C36).46C22, C12.57(C22×S3), C22.6(C9⋊D4), C12.113(C3⋊D4), (C9×C4○D4)⋊1C2, C2.24(C2×C9⋊D4), (C3×C4○D4).12S3, (C2×C6).8(C3⋊D4), C6.108(C2×C3⋊D4), SmallGroup(288,160)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D18
G = < a,b,c,d | a4=b2=c18=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, dbd=a-1b, dcd=c-1 >
Subgroups: 524 in 102 conjugacy classes, 38 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C9, C12, C12, D6, C2×C6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, D9, C18, C18, C3⋊C8, D12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C8⋊C22, C36, C36, D18, C2×C18, C2×C18, C4.Dic3, D4⋊S3, Q8⋊2S3, C2×D12, C3×C4○D4, C9⋊C8, D36, D36, C2×C36, C2×C36, D4×C9, D4×C9, Q8×C9, C22×D9, D4⋊D6, C4.Dic9, D4⋊D9, Q8⋊2D9, C2×D36, C9×C4○D4, D4⋊D18
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊D4, C22×S3, C8⋊C22, D18, C2×C3⋊D4, C9⋊D4, C22×D9, D4⋊D6, C2×C9⋊D4, D4⋊D18
►On 72 points
(1 34 15 26)(2 35 16 27)(3 36 17 19)(4 28 18 20)(5 29 10 21)(6 30 11 22)(7 31 12 23)(8 32 13 24)(9 33 14 25)(37 57 46 66)(38 58 47 67)(39 59 48 68)(40 60 49 69)(41 61 50 70)(42 62 51 71)(43 63 52 72)(44 64 53 55)(45 65 54 56)
(1 66)(2 58)(3 68)(4 60)(5 70)(6 62)(7 72)(8 64)(9 56)(10 61)(11 71)(12 63)(13 55)(14 65)(15 57)(16 67)(17 59)(18 69)(19 39)(20 49)(21 41)(22 51)(23 43)(24 53)(25 45)(26 37)(27 47)(28 40)(29 50)(30 42)(31 52)(32 44)(33 54)(34 46)(35 38)(36 48)
(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 9)(2 8)(3 7)(4 6)(11 18)(12 17)(13 16)(14 15)(19 31)(20 30)(21 29)(22 28)(23 36)(24 35)(25 34)(26 33)(27 32)(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)
G:=sub<Sym(72)| (1,34,15,26)(2,35,16,27)(3,36,17,19)(4,28,18,20)(5,29,10,21)(6,30,11,22)(7,31,12,23)(8,32,13,24)(9,33,14,25)(37,57,46,66)(38,58,47,67)(39,59,48,68)(40,60,49,69)(41,61,50,70)(42,62,51,71)(43,63,52,72)(44,64,53,55)(45,65,54,56), (1,66)(2,58)(3,68)(4,60)(5,70)(6,62)(7,72)(8,64)(9,56)(10,61)(11,71)(12,63)(13,55)(14,65)(15,57)(16,67)(17,59)(18,69)(19,39)(20,49)(21,41)(22,51)(23,43)(24,53)(25,45)(26,37)(27,47)(28,40)(29,50)(30,42)(31,52)(32,44)(33,54)(34,46)(35,38)(36,48), (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), (1,9)(2,8)(3,7)(4,6)(11,18)(12,17)(13,16)(14,15)(19,31)(20,30)(21,29)(22,28)(23,36)(24,35)(25,34)(26,33)(27,32)(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)>;
G:=Group( (1,34,15,26)(2,35,16,27)(3,36,17,19)(4,28,18,20)(5,29,10,21)(6,30,11,22)(7,31,12,23)(8,32,13,24)(9,33,14,25)(37,57,46,66)(38,58,47,67)(39,59,48,68)(40,60,49,69)(41,61,50,70)(42,62,51,71)(43,63,52,72)(44,64,53,55)(45,65,54,56), (1,66)(2,58)(3,68)(4,60)(5,70)(6,62)(7,72)(8,64)(9,56)(10,61)(11,71)(12,63)(13,55)(14,65)(15,57)(16,67)(17,59)(18,69)(19,39)(20,49)(21,41)(22,51)(23,43)(24,53)(25,45)(26,37)(27,47)(28,40)(29,50)(30,42)(31,52)(32,44)(33,54)(34,46)(35,38)(36,48), (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), (1,9)(2,8)(3,7)(4,6)(11,18)(12,17)(13,16)(14,15)(19,31)(20,30)(21,29)(22,28)(23,36)(24,35)(25,34)(26,33)(27,32)(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) );
G=PermutationGroup([[(1,34,15,26),(2,35,16,27),(3,36,17,19),(4,28,18,20),(5,29,10,21),(6,30,11,22),(7,31,12,23),(8,32,13,24),(9,33,14,25),(37,57,46,66),(38,58,47,67),(39,59,48,68),(40,60,49,69),(41,61,50,70),(42,62,51,71),(43,63,52,72),(44,64,53,55),(45,65,54,56)], [(1,66),(2,58),(3,68),(4,60),(5,70),(6,62),(7,72),(8,64),(9,56),(10,61),(11,71),(12,63),(13,55),(14,65),(15,57),(16,67),(17,59),(18,69),(19,39),(20,49),(21,41),(22,51),(23,43),(24,53),(25,45),(26,37),(27,47),(28,40),(29,50),(30,42),(31,52),(32,44),(33,54),(34,46),(35,38),(36,48)], [(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)], [(1,9),(2,8),(3,7),(4,6),(11,18),(12,17),(13,16),(14,15),(19,31),(20,30),(21,29),(22,28),(23,36),(24,35),(25,34),(26,33),(27,32),(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)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 18A | 18B | 18C | 18D | ··· | 18L | 36A | ··· | 36F | 36G | ··· | 36O |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 4 | 36 | 36 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
51 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D9 | C3⋊D4 | C3⋊D4 | D18 | D18 | D18 | C9⋊D4 | C9⋊D4 | C8⋊C22 | D4⋊D6 | D4⋊D18 |
| kernel | D4⋊D18 | C4.Dic9 | D4⋊D9 | Q8⋊2D9 | C2×D36 | C9×C4○D4 | C3×C4○D4 | C36 | C2×C18 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C12 | C2×C6 | C2×C4 | D4 | Q8 | C4 | C22 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 3 | 3 | 6 | 6 | 1 | 2 | 6 |
Matrix representation of D4⋊D18 ►in GL4(𝔽73) generated by
| 66 | 59 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 48 | 20 | 7 | 14 |
| 53 | 28 | 59 | 66 |
| 31 | 36 | 4 | 31 |
| 37 | 68 | 42 | 46 |
| 50 | 38 | 42 | 37 |
| 35 | 12 | 36 | 5 |
| 28 | 70 | 0 | 0 |
| 3 | 31 | 0 | 0 |
| 25 | 45 | 45 | 3 |
| 28 | 53 | 70 | 42 |
| 45 | 42 | 0 | 0 |
| 70 | 28 | 0 | 0 |
| 0 | 0 | 66 | 59 |
| 0 | 0 | 66 | 7 |
G:=sub<GL(4,GF(73))| [66,14,48,53,59,7,20,28,0,0,7,59,0,0,14,66],[31,37,50,35,36,68,38,12,4,42,42,36,31,46,37,5],[28,3,25,28,70,31,45,53,0,0,45,70,0,0,3,42],[45,70,0,0,42,28,0,0,0,0,66,66,0,0,59,7] >;
D4⋊D18 in GAP, Magma, Sage, TeX
D_4\rtimes D_{18} % in TeX
G:=Group("D4:D18"); // GroupNames label
G:=SmallGroup(288,160);
// by ID
G=gap.SmallGroup(288,160);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,219,675,185,80,6725,292,9414]);
// 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,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations