non-abelian, supersoluble, monomial
Aliases: C62.16D6, (D4xHe3):6C2, He3:7D4:4C2, He3:4Q8:5C2, (C3xC12).32D6, (D4xC32):6S3, He3:10(C4oD4), D4:2(He3:C2), C32:6(D4:2S3), (C2xHe3).34C23, (C4xHe3).24C22, C3.2(C12.D6), He3:3C4.19C22, (C22xHe3).14C22, C12.49(C2xC3:S3), (C2xHe3:3C4):7C2, (C4xHe3:C2):5C2, C6.66(C22xC3:S3), C4.5(C2xHe3:C2), (C3xD4).10(C3:S3), (C3xC6).44(C22xS3), C2.7(C22xHe3:C2), C22.1(C2xHe3:C2), (C2xHe3:C2).18C22, (C2xC6).10(C2xC3:S3), SmallGroup(432,391)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.16D6
G = < a,b,c,d | a6=b6=1, c6=d2=b3, ab=ba, cac-1=ab-1, dad-1=a-1b4, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 749 in 220 conjugacy classes, 53 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, D4, D4, Q8, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C4oD4, C3xS3, C3xC6, C3xC6, Dic6, C4xS3, C2xDic3, C3:D4, C2xC12, C3xD4, C3xD4, C3xQ8, He3, C3xDic3, C3xC12, S3xC6, C62, D4:2S3, C3xC4oD4, He3:C2, C2xHe3, C2xHe3, C3xDic6, S3xC12, C6xDic3, C3xC3:D4, D4xC32, He3:3C4, He3:3C4, C4xHe3, C2xHe3:C2, C22xHe3, C3xD4:2S3, He3:4Q8, C4xHe3:C2, C2xHe3:3C4, He3:7D4, D4xHe3, C62.16D6
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C3:S3, C22xS3, C2xC3:S3, D4:2S3, He3:C2, C22xC3:S3, C2xHe3:C2, C12.D6, C22xHe3:C2, C62.16D6
►On 72 points
(1 69 33 47 29 61)(2 46 60 42 6 14)(3 17 11 37 57 45)(4 66 36 44 32 70)(5 43 51 39 9 23)(7 63 27 41 35 67)(8 40 54 48 12 20)(10 72 30 38 26 64)(13 31 21 53 71 49)(15 59 65 55 19 25)(16 28 24 50 62 58)(18 56 68 52 22 34)
(1 31 51 7 25 57)(2 32 52 8 26 58)(3 33 53 9 27 59)(4 34 54 10 28 60)(5 35 55 11 29 49)(6 36 56 12 30 50)(13 43 67 19 37 61)(14 44 68 20 38 62)(15 45 69 21 39 63)(16 46 70 22 40 64)(17 47 71 23 41 65)(18 48 72 24 42 66)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 20 19 14)(15 18 21 24)(16 23 22 17)(25 30 31 36)(26 35 32 29)(27 28 33 34)(37 44 43 38)(39 42 45 48)(40 47 46 41)(49 58 55 52)(50 51 56 57)(53 54 59 60)(61 68 67 62)(63 66 69 72)(64 71 70 65)
G:=sub<Sym(72)| (1,69,33,47,29,61)(2,46,60,42,6,14)(3,17,11,37,57,45)(4,66,36,44,32,70)(5,43,51,39,9,23)(7,63,27,41,35,67)(8,40,54,48,12,20)(10,72,30,38,26,64)(13,31,21,53,71,49)(15,59,65,55,19,25)(16,28,24,50,62,58)(18,56,68,52,22,34), (1,31,51,7,25,57)(2,32,52,8,26,58)(3,33,53,9,27,59)(4,34,54,10,28,60)(5,35,55,11,29,49)(6,36,56,12,30,50)(13,43,67,19,37,61)(14,44,68,20,38,62)(15,45,69,21,39,63)(16,46,70,22,40,64)(17,47,71,23,41,65)(18,48,72,24,42,66), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,20,19,14)(15,18,21,24)(16,23,22,17)(25,30,31,36)(26,35,32,29)(27,28,33,34)(37,44,43,38)(39,42,45,48)(40,47,46,41)(49,58,55,52)(50,51,56,57)(53,54,59,60)(61,68,67,62)(63,66,69,72)(64,71,70,65)>;
G:=Group( (1,69,33,47,29,61)(2,46,60,42,6,14)(3,17,11,37,57,45)(4,66,36,44,32,70)(5,43,51,39,9,23)(7,63,27,41,35,67)(8,40,54,48,12,20)(10,72,30,38,26,64)(13,31,21,53,71,49)(15,59,65,55,19,25)(16,28,24,50,62,58)(18,56,68,52,22,34), (1,31,51,7,25,57)(2,32,52,8,26,58)(3,33,53,9,27,59)(4,34,54,10,28,60)(5,35,55,11,29,49)(6,36,56,12,30,50)(13,43,67,19,37,61)(14,44,68,20,38,62)(15,45,69,21,39,63)(16,46,70,22,40,64)(17,47,71,23,41,65)(18,48,72,24,42,66), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,20,19,14)(15,18,21,24)(16,23,22,17)(25,30,31,36)(26,35,32,29)(27,28,33,34)(37,44,43,38)(39,42,45,48)(40,47,46,41)(49,58,55,52)(50,51,56,57)(53,54,59,60)(61,68,67,62)(63,66,69,72)(64,71,70,65) );
G=PermutationGroup([[(1,69,33,47,29,61),(2,46,60,42,6,14),(3,17,11,37,57,45),(4,66,36,44,32,70),(5,43,51,39,9,23),(7,63,27,41,35,67),(8,40,54,48,12,20),(10,72,30,38,26,64),(13,31,21,53,71,49),(15,59,65,55,19,25),(16,28,24,50,62,58),(18,56,68,52,22,34)], [(1,31,51,7,25,57),(2,32,52,8,26,58),(3,33,53,9,27,59),(4,34,54,10,28,60),(5,35,55,11,29,49),(6,36,56,12,30,50),(13,43,67,19,37,61),(14,44,68,20,38,62),(15,45,69,21,39,63),(16,46,70,22,40,64),(17,47,71,23,41,65),(18,48,72,24,42,66)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,20,19,14),(15,18,21,24),(16,23,22,17),(25,30,31,36),(26,35,32,29),(27,28,33,34),(37,44,43,38),(39,42,45,48),(40,47,46,41),(49,58,55,52),(50,51,56,57),(53,54,59,60),(61,68,67,62),(63,66,69,72),(64,71,70,65)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | ··· | 6R | 6S | 6T | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 12M | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 2 | 18 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 9 | 9 | 18 | 18 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 18 | 18 | 2 | 2 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 6 |
| type | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4oD4 | He3:C2 | C2xHe3:C2 | C2xHe3:C2 | D4:2S3 | C62.16D6 |
| kernel | C62.16D6 | He3:4Q8 | C4xHe3:C2 | C2xHe3:3C4 | He3:7D4 | D4xHe3 | D4xC32 | C3xC12 | C62 | He3 | D4 | C4 | C22 | C32 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 4 | 8 | 2 | 4 | 4 | 8 | 4 | 4 |
Matrix representation of C62.16D6 ►in GL5(F13)
| 0 | 8 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 10 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(5,GF(13))| [0,5,0,0,0,8,0,0,0,0,0,0,0,0,10,0,0,1,0,0,0,0,0,9,0],[12,0,0,0,0,0,12,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[5,0,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,1,0],[5,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0] >;
C62.16D6 in GAP, Magma, Sage, TeX
C_6^2._{16}D_6 % in TeX
G:=Group("C6^2.16D6"); // GroupNames label
G:=SmallGroup(432,391);
// by ID
G=gap.SmallGroup(432,391);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,1124,4037,537]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^6=d^2=b^3,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b^4,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations