direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2xC18.D6, Dic9:6D6, Dic3:6D18, C62.63D6, C6:1(C4xD9), C18:1(C4xS3), (C2xDic3):5D9, (C6xDic9):5C2, (C2xDic9):5S3, (C2xC18).18D6, (C2xC6).17D18, (Dic3xC18):8C2, C22.13(S3xD9), C6.18(C22xD9), C18.18(C22xS3), (C3xC18).18C23, (C6xC18).12C22, (C6xDic3).16S3, (C3xDic3).41D6, C6.6(C6.D6), (C3xDic9):5C22, (C9xDic3):7C22, C9:2(S3xC2xC4), C3:2(C2xC4xD9), (C2xC9:S3):3C4, C9:S3:2(C2xC4), C2.3(C2xS3xD9), C6.37(C2xS32), (C2xC6).24S32, (C3xC18):2(C2xC4), (C3xC9):3(C22xC4), C32.4(S3xC2xC4), (C3xC6).43(C4xS3), (C22xC9:S3).3C2, C3.1(C2xC6.D6), (C2xC9:S3).11C22, (C3xC6).86(C22xS3), SmallGroup(432,306)
Series: Derived ►Chief ►Lower central ►Upper central
| C3xC9 — C2xC18.D6 |
Generators and relations for C2xC18.D6
G = < a,b,c,d | a2=b18=d2=1, c6=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c5 >
Subgroups: 1260 in 194 conjugacy classes, 61 normal (25 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2xC4, C23, C9, C9, C32, Dic3, Dic3, C12, D6, C2xC6, C2xC6, C22xC4, D9, C18, C18, C18, C3:S3, C3xC6, C3xC6, C4xS3, C2xDic3, C2xDic3, C2xC12, C22xS3, C3xC9, Dic9, C36, D18, C2xC18, C2xC18, C3xDic3, C3xDic3, C2xC3:S3, C62, S3xC2xC4, C9:S3, C3xC18, C3xC18, C4xD9, C2xDic9, C2xC36, C22xD9, C6.D6, C6xDic3, C6xDic3, C22xC3:S3, C3xDic9, C9xDic3, C2xC9:S3, C6xC18, C2xC4xD9, C2xC6.D6, C18.D6, C6xDic9, Dic3xC18, C22xC9:S3, C2xC18.D6
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, C22xC4, D9, C4xS3, C22xS3, D18, S32, S3xC2xC4, C4xD9, C22xD9, C6.D6, C2xS32, S3xD9, C2xC4xD9, C2xC6.D6, C18.D6, C2xS3xD9, C2xC18.D6
►On 72 points
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(37 72)(38 55)(39 56)(40 57)(41 58)(42 59)(43 60)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)
(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 59 16 62 13 65 10 68 7 71 4 56)(2 58 17 61 14 64 11 67 8 70 5 55)(3 57 18 60 15 63 12 66 9 69 6 72)(19 50 34 53 31 38 28 41 25 44 22 47)(20 49 35 52 32 37 29 40 26 43 23 46)(21 48 36 51 33 54 30 39 27 42 24 45)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 18)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(33 36)(34 35)(37 38)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)
G:=sub<Sym(72)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(37,72)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71), (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), (1,59,16,62,13,65,10,68,7,71,4,56)(2,58,17,61,14,64,11,67,8,70,5,55)(3,57,18,60,15,63,12,66,9,69,6,72)(19,50,34,53,31,38,28,41,25,44,22,47)(20,49,35,52,32,37,29,40,26,43,23,46)(21,48,36,51,33,54,30,39,27,42,24,45), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,18)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(33,36)(34,35)(37,38)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(37,72)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71), (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), (1,59,16,62,13,65,10,68,7,71,4,56)(2,58,17,61,14,64,11,67,8,70,5,55)(3,57,18,60,15,63,12,66,9,69,6,72)(19,50,34,53,31,38,28,41,25,44,22,47)(20,49,35,52,32,37,29,40,26,43,23,46)(21,48,36,51,33,54,30,39,27,42,24,45), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,18)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(33,36)(34,35)(37,38)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64) );
G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26),(37,72),(38,55),(39,56),(40,57),(41,58),(42,59),(43,60),(44,61),(45,62),(46,63),(47,64),(48,65),(49,66),(50,67),(51,68),(52,69),(53,70),(54,71)], [(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)], [(1,59,16,62,13,65,10,68,7,71,4,56),(2,58,17,61,14,64,11,67,8,70,5,55),(3,57,18,60,15,63,12,66,9,69,6,72),(19,50,34,53,31,38,28,41,25,44,22,47),(20,49,35,52,32,37,29,40,26,43,23,46),(21,48,36,51,33,54,30,39,27,42,24,45)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,18),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(25,26),(33,36),(34,35),(37,38),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(55,72),(56,71),(57,70),(58,69),(59,68),(60,67),(61,66),(62,65),(63,64)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | ··· | 18I | 18J | ··· | 18R | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 27 | 27 | 27 | 27 | 2 | 2 | 4 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | S3 | D6 | D6 | D6 | D6 | D9 | C4xS3 | C4xS3 | D18 | D18 | C4xD9 | S32 | C6.D6 | C2xS32 | S3xD9 | C18.D6 | C2xS3xD9 |
| kernel | C2xC18.D6 | C18.D6 | C6xDic9 | Dic3xC18 | C22xC9:S3 | C2xC9:S3 | C2xDic9 | C6xDic3 | Dic9 | C2xC18 | C3xDic3 | C62 | C2xDic3 | C18 | C3xC6 | Dic3 | C2xC6 | C6 | C2xC6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 4 | 4 | 6 | 3 | 12 | 1 | 2 | 1 | 3 | 6 | 3 |
Matrix representation of C2xC18.D6 ►in GL4(F37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 26 | 31 |
| 0 | 0 | 6 | 20 |
| 6 | 31 | 0 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 0 | 1 | 36 |
| 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 |
| 1 | 36 | 0 | 0 |
| 0 | 0 | 1 | 36 |
| 0 | 0 | 0 | 36 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,36,0,0,0,0,26,6,0,0,31,20],[6,6,0,0,31,0,0,0,0,0,1,0,0,0,36,36],[1,1,0,0,0,36,0,0,0,0,1,0,0,0,36,36] >;
C2xC18.D6 in GAP, Magma, Sage, TeX
C_2\times C_{18}.D_6 % in TeX
G:=Group("C2xC18.D6"); // GroupNames label
G:=SmallGroup(432,306);
// by ID
G=gap.SmallGroup(432,306);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^18=d^2=1,c^6=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^5>;
// generators/relations