metabelian, supersoluble, monomial, A-group
Aliases: C6.3D30, C30.23D6, Dic3⋊2D15, Dic15⋊5S3, C10.3S32, C3⋊D15⋊6C4, C3⋊1(C4×D15), C6.3(S3×D5), C15⋊10(C4×S3), C32⋊3(C4×D5), C2.3(S3×D15), (C3×C6).3D10, (C5×Dic3)⋊2S3, (C3×Dic3)⋊3D5, C5⋊2(C6.D6), C3⋊1(D30.C2), (C3×Dic15)⋊6C2, (Dic3×C15)⋊3C2, (C3×C30).17C22, (C3×C15)⋊21(C2×C4), (C2×C3⋊D15).3C2, SmallGroup(360,79)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — C6.D30 |
Generators and relations for C6.D30
G = < a,b,c | a6=c2=1, b30=a3, bab-1=cac=a-1, cbc=b29 >
Subgroups: 580 in 74 conjugacy classes, 26 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, C2×C4, C32, D5, C10, Dic3, Dic3, C12, D6, C15, C15, C3⋊S3, C3×C6, Dic5, C20, D10, C4×S3, D15, C30, C30, C3×Dic3, C3×Dic3, C2×C3⋊S3, C4×D5, C3×C15, C5×Dic3, C3×Dic5, Dic15, C60, D30, C6.D6, C3⋊D15, C3×C30, D30.C2, C4×D15, Dic3×C15, C3×Dic15, C2×C3⋊D15, C6.D30
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, D6, D10, C4×S3, D15, S32, C4×D5, S3×D5, D30, C6.D6, D30.C2, C4×D15, S3×D15, C6.D30
►On 60 points
(1 11 21 31 41 51)(2 52 42 32 22 12)(3 13 23 33 43 53)(4 54 44 34 24 14)(5 15 25 35 45 55)(6 56 46 36 26 16)(7 17 27 37 47 57)(8 58 48 38 28 18)(9 19 29 39 49 59)(10 60 50 40 30 20)
(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)
(1 45)(2 14)(3 43)(4 12)(5 41)(6 10)(7 39)(9 37)(11 35)(13 33)(15 31)(16 60)(17 29)(18 58)(19 27)(20 56)(21 25)(22 54)(24 52)(26 50)(28 48)(30 46)(32 44)(34 42)(36 40)(47 59)(49 57)(51 55)
G:=sub<Sym(60)| (1,11,21,31,41,51)(2,52,42,32,22,12)(3,13,23,33,43,53)(4,54,44,34,24,14)(5,15,25,35,45,55)(6,56,46,36,26,16)(7,17,27,37,47,57)(8,58,48,38,28,18)(9,19,29,39,49,59)(10,60,50,40,30,20), (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), (1,45)(2,14)(3,43)(4,12)(5,41)(6,10)(7,39)(9,37)(11,35)(13,33)(15,31)(16,60)(17,29)(18,58)(19,27)(20,56)(21,25)(22,54)(24,52)(26,50)(28,48)(30,46)(32,44)(34,42)(36,40)(47,59)(49,57)(51,55)>;
G:=Group( (1,11,21,31,41,51)(2,52,42,32,22,12)(3,13,23,33,43,53)(4,54,44,34,24,14)(5,15,25,35,45,55)(6,56,46,36,26,16)(7,17,27,37,47,57)(8,58,48,38,28,18)(9,19,29,39,49,59)(10,60,50,40,30,20), (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), (1,45)(2,14)(3,43)(4,12)(5,41)(6,10)(7,39)(9,37)(11,35)(13,33)(15,31)(16,60)(17,29)(18,58)(19,27)(20,56)(21,25)(22,54)(24,52)(26,50)(28,48)(30,46)(32,44)(34,42)(36,40)(47,59)(49,57)(51,55) );
G=PermutationGroup([[(1,11,21,31,41,51),(2,52,42,32,22,12),(3,13,23,33,43,53),(4,54,44,34,24,14),(5,15,25,35,45,55),(6,56,46,36,26,16),(7,17,27,37,47,57),(8,58,48,38,28,18),(9,19,29,39,49,59),(10,60,50,40,30,20)], [(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)], [(1,45),(2,14),(3,43),(4,12),(5,41),(6,10),(7,39),(9,37),(11,35),(13,33),(15,31),(16,60),(17,29),(18,58),(19,27),(20,56),(21,25),(22,54),(24,52),(26,50),(28,48),(30,46),(32,44),(34,42),(36,40),(47,59),(49,57),(51,55)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 15E | ··· | 15J | 20A | 20B | 20C | 20D | 30A | 30B | 30C | 30D | 30E | ··· | 30J | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 45 | 45 | 2 | 2 | 4 | 3 | 3 | 15 | 15 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 6 | 6 | 30 | 30 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | D5 | D6 | D10 | C4×S3 | D15 | C4×D5 | D30 | C4×D15 | S32 | S3×D5 | C6.D6 | D30.C2 | S3×D15 | C6.D30 |
| kernel | C6.D30 | Dic3×C15 | C3×Dic15 | C2×C3⋊D15 | C3⋊D15 | C5×Dic3 | Dic15 | C3×Dic3 | C30 | C3×C6 | C15 | Dic3 | C32 | C6 | C3 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | 2 | 1 | 2 | 4 | 4 |
Matrix representation of C6.D30 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 17 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 11 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 17 | 60 | 0 | 0 | 0 | 0 |
| 44 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[17,1,0,0,0,0,60,0,0,0,0,0,0,0,0,11,0,0,0,0,50,50,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[17,44,0,0,0,0,60,44,0,0,0,0,0,0,1,0,0,0,0,0,60,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C6.D30 in GAP, Magma, Sage, TeX
C_6.D_{30} % in TeX
G:=Group("C6.D30"); // GroupNames label
G:=SmallGroup(360,79);
// by ID
G=gap.SmallGroup(360,79);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,31,201,1444,10373]);
// Polycyclic
G:=Group<a,b,c|a^6=c^2=1,b^30=a^3,b*a*b^-1=c*a*c=a^-1,c*b*c=b^29>;
// generators/relations