direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C18, C23⋊2C18, C36⋊4C22, C18.11C23, C4⋊(C2×C18), C3.(C6×D4), (C6×D4).C3, (C2×C4)⋊2C18, (C2×C36)⋊6C2, (C3×D4).5C6, C6.17(C3×D4), (C2×C12).10C6, C12.20(C2×C6), C22⋊2(C2×C18), (C2×C18)⋊2C22, (C22×C18)⋊1C2, (C22×C6).6C6, C6.11(C22×C6), C2.1(C22×C18), (C2×C18)○(C6×D4), (C2×C6).3(C2×C6), SmallGroup(144,48)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C18
G = < a,b,c | a18=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 105 in 81 conjugacy classes, 57 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, D4, C23, C9, C12, C2×C6, C2×C6, C2×C6, C2×D4, C18, C18, C18, C2×C12, C3×D4, C22×C6, C36, C2×C18, C2×C18, C2×C18, C6×D4, C2×C36, D4×C9, C22×C18, D4×C18
Quotients: C1, C2, C3, C22, C6, D4, C23, C9, C2×C6, C2×D4, C18, C3×D4, C22×C6, C2×C18, C6×D4, D4×C9, C22×C18, D4×C18
►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 37 59 31)(2 38 60 32)(3 39 61 33)(4 40 62 34)(5 41 63 35)(6 42 64 36)(7 43 65 19)(8 44 66 20)(9 45 67 21)(10 46 68 22)(11 47 69 23)(12 48 70 24)(13 49 71 25)(14 50 72 26)(15 51 55 27)(16 52 56 28)(17 53 57 29)(18 54 58 30)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 55)
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), (1,37,59,31)(2,38,60,32)(3,39,61,33)(4,40,62,34)(5,41,63,35)(6,42,64,36)(7,43,65,19)(8,44,66,20)(9,45,67,21)(10,46,68,22)(11,47,69,23)(12,48,70,24)(13,49,71,25)(14,50,72,26)(15,51,55,27)(16,52,56,28)(17,53,57,29)(18,54,58,30), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,55)>;
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), (1,37,59,31)(2,38,60,32)(3,39,61,33)(4,40,62,34)(5,41,63,35)(6,42,64,36)(7,43,65,19)(8,44,66,20)(9,45,67,21)(10,46,68,22)(11,47,69,23)(12,48,70,24)(13,49,71,25)(14,50,72,26)(15,51,55,27)(16,52,56,28)(17,53,57,29)(18,54,58,30), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,55) );
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)], [(1,37,59,31),(2,38,60,32),(3,39,61,33),(4,40,62,34),(5,41,63,35),(6,42,64,36),(7,43,65,19),(8,44,66,20),(9,45,67,21),(10,46,68,22),(11,47,69,23),(12,48,70,24),(13,49,71,25),(14,50,72,26),(15,51,55,27),(16,52,56,28),(17,53,57,29),(18,54,58,30)], [(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,55)]])
D4×C18 is a maximal subgroup of
C36.D4 D4⋊Dic9 C23⋊2Dic9 D36⋊6C22 C23.23D18 C36.17D4 C23⋊2D18 C36⋊2D4 Dic9⋊D4 C36⋊D4 D4⋊6D18
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 6A | ··· | 6F | 6G | ··· | 6N | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 18A | ··· | 18R | 18S | ··· | 18AP | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | D4 | C3×D4 | D4×C9 |
| kernel | D4×C18 | C2×C36 | D4×C9 | C22×C18 | C6×D4 | C2×C12 | C3×D4 | C22×C6 | C2×D4 | C2×C4 | D4 | C23 | C18 | C6 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 8 | 4 | 6 | 6 | 24 | 12 | 2 | 4 | 12 |
Matrix representation of D4×C18 ►in GL3(𝔽37) generated by
| 11 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 1 | 0 | 0 |
| 0 | 1 | 14 |
| 0 | 21 | 36 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 21 | 36 |
G:=sub<GL(3,GF(37))| [11,0,0,0,12,0,0,0,12],[1,0,0,0,1,21,0,14,36],[1,0,0,0,1,21,0,0,36] >;
D4×C18 in GAP, Magma, Sage, TeX
D_4\times C_{18} % in TeX
G:=Group("D4xC18"); // GroupNames label
G:=SmallGroup(144,48);
// by ID
G=gap.SmallGroup(144,48);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-3,313,165]);
// Polycyclic
G:=Group<a,b,c|a^18=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations