metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊6D18, C23⋊3D18, D36⋊8C22, C18.7C24, C9⋊12+ 1+4, C36.21C23, D18.3C23, Dic18⋊8C22, Dic9.4C23, (D4×D9)⋊4C2, (C2×D4)⋊7D9, (C2×C4)⋊3D18, (D4×C18)⋊7C2, D4⋊2D9⋊4C2, C3.(D4⋊6D6), (C2×C36)⋊3C22, (C6×D4).14S3, (C3×D4).37D6, (C4×D9)⋊1C22, (D4×C9)⋊7C22, C9⋊D4⋊3C22, D36⋊5C2⋊5C2, C2.8(C23×D9), (C2×C12).100D6, (C2×C18).2C23, C6.44(S3×C23), C4.21(C22×D9), (C22×C6).60D6, C12.61(C22×S3), (C22×C18)⋊5C22, (C2×Dic9)⋊4C22, (C22×D9)⋊3C22, C22.6(C22×D9), (C2×C9⋊D4)⋊11C2, (C2×C6).223(C22×S3), SmallGroup(288,358)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊6D18
G = < a,b,c,d | a4=b2=c18=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 1080 in 249 conjugacy classes, 102 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C2×D4, C2×D4, C4○D4, D9, C18, C18, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, 2+ 1+4, Dic9, C36, D18, D18, C2×C18, C2×C18, C2×C18, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C6×D4, Dic18, C4×D9, D36, C2×Dic9, C9⋊D4, C2×C36, D4×C9, C22×D9, C22×C18, D4⋊6D6, D36⋊5C2, D4×D9, D4⋊2D9, C2×C9⋊D4, D4×C18, D4⋊6D18
Quotients: C1, C2, C22, S3, C23, D6, C24, D9, C22×S3, 2+ 1+4, D18, S3×C23, C22×D9, D4⋊6D6, C23×D9, D4⋊6D18
►On 72 points
(1 62 17 71)(2 72 18 63)(3 64 10 55)(4 56 11 65)(5 66 12 57)(6 58 13 67)(7 68 14 59)(8 60 15 69)(9 70 16 61)(19 42 29 51)(20 52 30 43)(21 44 31 53)(22 54 32 45)(23 46 33 37)(24 38 34 47)(25 48 35 39)(26 40 36 49)(27 50 28 41)
(1 39)(2 49)(3 41)(4 51)(5 43)(6 53)(7 45)(8 37)(9 47)(10 50)(11 42)(12 52)(13 44)(14 54)(15 46)(16 38)(17 48)(18 40)(19 65)(20 57)(21 67)(22 59)(23 69)(24 61)(25 71)(26 63)(27 55)(28 64)(29 56)(30 66)(31 58)(32 68)(33 60)(34 70)(35 62)(36 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 9)(2 8)(3 7)(4 6)(10 14)(11 13)(15 18)(16 17)(19 31)(20 30)(21 29)(22 28)(23 36)(24 35)(25 34)(26 33)(27 32)(37 40)(38 39)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 59)(56 58)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)
G:=sub<Sym(72)| (1,62,17,71)(2,72,18,63)(3,64,10,55)(4,56,11,65)(5,66,12,57)(6,58,13,67)(7,68,14,59)(8,60,15,69)(9,70,16,61)(19,42,29,51)(20,52,30,43)(21,44,31,53)(22,54,32,45)(23,46,33,37)(24,38,34,47)(25,48,35,39)(26,40,36,49)(27,50,28,41), (1,39)(2,49)(3,41)(4,51)(5,43)(6,53)(7,45)(8,37)(9,47)(10,50)(11,42)(12,52)(13,44)(14,54)(15,46)(16,38)(17,48)(18,40)(19,65)(20,57)(21,67)(22,59)(23,69)(24,61)(25,71)(26,63)(27,55)(28,64)(29,56)(30,66)(31,58)(32,68)(33,60)(34,70)(35,62)(36,72), (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), (1,9)(2,8)(3,7)(4,6)(10,14)(11,13)(15,18)(16,17)(19,31)(20,30)(21,29)(22,28)(23,36)(24,35)(25,34)(26,33)(27,32)(37,40)(38,39)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,59)(56,58)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67)>;
G:=Group( (1,62,17,71)(2,72,18,63)(3,64,10,55)(4,56,11,65)(5,66,12,57)(6,58,13,67)(7,68,14,59)(8,60,15,69)(9,70,16,61)(19,42,29,51)(20,52,30,43)(21,44,31,53)(22,54,32,45)(23,46,33,37)(24,38,34,47)(25,48,35,39)(26,40,36,49)(27,50,28,41), (1,39)(2,49)(3,41)(4,51)(5,43)(6,53)(7,45)(8,37)(9,47)(10,50)(11,42)(12,52)(13,44)(14,54)(15,46)(16,38)(17,48)(18,40)(19,65)(20,57)(21,67)(22,59)(23,69)(24,61)(25,71)(26,63)(27,55)(28,64)(29,56)(30,66)(31,58)(32,68)(33,60)(34,70)(35,62)(36,72), (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), (1,9)(2,8)(3,7)(4,6)(10,14)(11,13)(15,18)(16,17)(19,31)(20,30)(21,29)(22,28)(23,36)(24,35)(25,34)(26,33)(27,32)(37,40)(38,39)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,59)(56,58)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67) );
G=PermutationGroup([[(1,62,17,71),(2,72,18,63),(3,64,10,55),(4,56,11,65),(5,66,12,57),(6,58,13,67),(7,68,14,59),(8,60,15,69),(9,70,16,61),(19,42,29,51),(20,52,30,43),(21,44,31,53),(22,54,32,45),(23,46,33,37),(24,38,34,47),(25,48,35,39),(26,40,36,49),(27,50,28,41)], [(1,39),(2,49),(3,41),(4,51),(5,43),(6,53),(7,45),(8,37),(9,47),(10,50),(11,42),(12,52),(13,44),(14,54),(15,46),(16,38),(17,48),(18,40),(19,65),(20,57),(21,67),(22,59),(23,69),(24,61),(25,71),(26,63),(27,55),(28,64),(29,56),(30,66),(31,58),(32,68),(33,60),(34,70),(35,62),(36,72)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,14),(11,13),(15,18),(16,17),(19,31),(20,30),(21,29),(22,28),(23,36),(24,35),(25,34),(26,33),(27,32),(37,40),(38,39),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,48),(55,59),(56,58),(60,72),(61,71),(62,70),(63,69),(64,68),(65,67)]])
57 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 12A | 12B | 18A | ··· | 18I | 18J | ··· | 18U | 36A | ··· | 36F |
| order | 1 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 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⋊6D6 | D4⋊6D18 |
| kernel | D4⋊6D18 | D36⋊5C2 | D4×D9 | D4⋊2D9 | C2×C9⋊D4 | D4×C18 | C6×D4 | C2×C12 | C3×D4 | C22×C6 | C2×D4 | C2×C4 | D4 | C23 | C9 | C3 | C1 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 4 | 2 | 3 | 3 | 12 | 6 | 1 | 2 | 6 |
Matrix representation of D4⋊6D18 ►in GL6(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 23 |
| 0 | 0 | 0 | 0 | 14 | 7 |
| 0 | 0 | 7 | 14 | 0 | 0 |
| 0 | 0 | 23 | 30 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 6 | 17 | 0 | 0 | 0 | 0 |
| 20 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 6 | 17 | 0 | 0 | 0 | 0 |
| 11 | 31 | 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 | 36 | 0 |
G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,23,0,0,0,0,14,30,0,0,30,14,0,0,0,0,23,7,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[6,20,0,0,0,0,17,26,0,0,0,0,0,0,0,36,0,0,0,0,1,36,0,0,0,0,0,0,0,1,0,0,0,0,36,1],[6,11,0,0,0,0,17,31,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,36,0] >;
D4⋊6D18 in GAP, Magma, Sage, TeX
D_4\rtimes_6D_{18} % in TeX
G:=Group("D4:6D18"); // GroupNames label
G:=SmallGroup(288,358);
// by ID
G=gap.SmallGroup(288,358);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^18=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations