direct product, metabelian, supersoluble, monomial
Aliases: C3×C18.D4, C62.122D6, C62.17Dic3, (C6×C18)⋊4C4, (C2×C6)⋊3Dic9, (C2×C18)⋊10C12, (C6×Dic9)⋊4C2, (C2×Dic9)⋊8C6, C18.27(C3×D4), (C3×C18).44D4, (C2×C6).50D18, C2.5(C6×Dic9), C23.3(C3×D9), C22.7(C6×D9), (C22×C6).5D9, C18.23(C2×C12), C6.35(C9⋊D4), (C2×C62).21S3, C6.18(C6×Dic3), C6.21(C2×Dic9), C22⋊3(C3×Dic9), (C22×C18).14C6, (C6×C18).36C22, C32.3(C6.D4), (C2×C6×C18).4C2, C9⋊5(C3×C22⋊C4), C2.3(C3×C9⋊D4), (C3×C9)⋊8(C22⋊C4), (C2×C6).46(S3×C6), C6.30(C3×C3⋊D4), (C3×C18).35(C2×C4), (C2×C18).27(C2×C6), (C3×C6).96(C3⋊D4), (C22×C6).24(C3×S3), (C3×C6).58(C2×Dic3), (C2×C6).18(C3×Dic3), C3.1(C3×C6.D4), SmallGroup(432,164)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C18.D4
G = < a,b,c,d | a3=b18=c4=1, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b9c-1 >
Subgroups: 342 in 134 conjugacy classes, 54 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C18, C18, C18, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C3×C9, Dic9, C2×C18, C2×C18, C2×C18, C3×Dic3, C62, C62, C62, C6.D4, C3×C22⋊C4, C3×C18, C3×C18, C3×C18, C2×Dic9, C22×C18, C22×C18, C6×Dic3, C2×C62, C3×Dic9, C6×C18, C6×C18, C6×C18, C18.D4, C3×C6.D4, C6×Dic9, C2×C6×C18, C3×C18.D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Dic3, C12, D6, C2×C6, C22⋊C4, D9, C3×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, Dic9, D18, C3×Dic3, S3×C6, C6.D4, C3×C22⋊C4, C3×D9, C2×Dic9, C9⋊D4, C6×Dic3, C3×C3⋊D4, C3×Dic9, C6×D9, C18.D4, C3×C6.D4, C6×Dic9, C3×C9⋊D4, C3×C18.D4
►On 72 points
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 61 67)(56 62 68)(57 63 69)(58 64 70)(59 65 71)(60 66 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 21 53 64)(2 20 54 63)(3 19 37 62)(4 36 38 61)(5 35 39 60)(6 34 40 59)(7 33 41 58)(8 32 42 57)(9 31 43 56)(10 30 44 55)(11 29 45 72)(12 28 46 71)(13 27 47 70)(14 26 48 69)(15 25 49 68)(16 24 50 67)(17 23 51 66)(18 22 52 65)
(1 55 10 64)(2 72 11 63)(3 71 12 62)(4 70 13 61)(5 69 14 60)(6 68 15 59)(7 67 16 58)(8 66 17 57)(9 65 18 56)(19 37 28 46)(20 54 29 45)(21 53 30 44)(22 52 31 43)(23 51 32 42)(24 50 33 41)(25 49 34 40)(26 48 35 39)(27 47 36 38)
G:=sub<Sym(72)| (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,61,67)(56,62,68)(57,63,69)(58,64,70)(59,65,71)(60,66,72), (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), (1,21,53,64)(2,20,54,63)(3,19,37,62)(4,36,38,61)(5,35,39,60)(6,34,40,59)(7,33,41,58)(8,32,42,57)(9,31,43,56)(10,30,44,55)(11,29,45,72)(12,28,46,71)(13,27,47,70)(14,26,48,69)(15,25,49,68)(16,24,50,67)(17,23,51,66)(18,22,52,65), (1,55,10,64)(2,72,11,63)(3,71,12,62)(4,70,13,61)(5,69,14,60)(6,68,15,59)(7,67,16,58)(8,66,17,57)(9,65,18,56)(19,37,28,46)(20,54,29,45)(21,53,30,44)(22,52,31,43)(23,51,32,42)(24,50,33,41)(25,49,34,40)(26,48,35,39)(27,47,36,38)>;
G:=Group( (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,61,67)(56,62,68)(57,63,69)(58,64,70)(59,65,71)(60,66,72), (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), (1,21,53,64)(2,20,54,63)(3,19,37,62)(4,36,38,61)(5,35,39,60)(6,34,40,59)(7,33,41,58)(8,32,42,57)(9,31,43,56)(10,30,44,55)(11,29,45,72)(12,28,46,71)(13,27,47,70)(14,26,48,69)(15,25,49,68)(16,24,50,67)(17,23,51,66)(18,22,52,65), (1,55,10,64)(2,72,11,63)(3,71,12,62)(4,70,13,61)(5,69,14,60)(6,68,15,59)(7,67,16,58)(8,66,17,57)(9,65,18,56)(19,37,28,46)(20,54,29,45)(21,53,30,44)(22,52,31,43)(23,51,32,42)(24,50,33,41)(25,49,34,40)(26,48,35,39)(27,47,36,38) );
G=PermutationGroup([[(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12),(19,25,31),(20,26,32),(21,27,33),(22,28,34),(23,29,35),(24,30,36),(37,49,43),(38,50,44),(39,51,45),(40,52,46),(41,53,47),(42,54,48),(55,61,67),(56,62,68),(57,63,69),(58,64,70),(59,65,71),(60,66,72)], [(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)], [(1,21,53,64),(2,20,54,63),(3,19,37,62),(4,36,38,61),(5,35,39,60),(6,34,40,59),(7,33,41,58),(8,32,42,57),(9,31,43,56),(10,30,44,55),(11,29,45,72),(12,28,46,71),(13,27,47,70),(14,26,48,69),(15,25,49,68),(16,24,50,67),(17,23,51,66),(18,22,52,65)], [(1,55,10,64),(2,72,11,63),(3,71,12,62),(4,70,13,61),(5,69,14,60),(6,68,15,59),(7,67,16,58),(8,66,17,57),(9,65,18,56),(19,37,28,46),(20,54,29,45),(21,53,30,44),(22,52,31,43),(23,51,32,42),(24,50,33,41),(25,49,34,40),(26,48,35,39),(27,47,36,38)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6AE | 9A | ··· | 9I | 12A | ··· | 12H | 18A | ··· | 18BK |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 18 | ··· | 18 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + | ||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | D4 | Dic3 | D6 | D9 | C3×S3 | C3×D4 | C3⋊D4 | Dic9 | D18 | C3×Dic3 | S3×C6 | C3×D9 | C9⋊D4 | C3×C3⋊D4 | C3×Dic9 | C6×D9 | C3×C9⋊D4 |
| kernel | C3×C18.D4 | C6×Dic9 | C2×C6×C18 | C18.D4 | C6×C18 | C2×Dic9 | C22×C18 | C2×C18 | C2×C62 | C3×C18 | C62 | C62 | C22×C6 | C22×C6 | C18 | C3×C6 | C2×C6 | C2×C6 | C2×C6 | C2×C6 | C23 | C6 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 2 | 2 | 1 | 3 | 2 | 4 | 4 | 6 | 3 | 4 | 2 | 6 | 12 | 8 | 12 | 6 | 24 |
Matrix representation of C3×C18.D4 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 26 | 0 |
| 0 | 0 | 0 | 26 |
| 10 | 0 | 0 | 0 |
| 5 | 26 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 4 |
| 31 | 3 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 0 |
| 31 | 3 | 0 | 0 |
| 13 | 6 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 0 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,26,0,0,0,0,26],[10,5,0,0,0,26,0,0,0,0,28,0,0,0,0,4],[31,0,0,0,3,6,0,0,0,0,0,36,0,0,1,0],[31,13,0,0,3,6,0,0,0,0,0,36,0,0,1,0] >;
C3×C18.D4 in GAP, Magma, Sage, TeX
C_3\times C_{18}.D_4 % in TeX
G:=Group("C3xC18.D4"); // GroupNames label
G:=SmallGroup(432,164);
// by ID
G=gap.SmallGroup(432,164);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^18=c^4=1,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^9*c^-1>;
// generators/relations