metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊2D6, D10⋊2D6, D20⋊4S3, D6⋊2D10, D15⋊1D4, D12⋊4D5, C12⋊2D10, C60⋊4C22, C30.14C23, Dic15⋊6C22, D30.13C22, C3⋊2(D4×D5), C5⋊2(S3×D4), C4⋊3(S3×D5), C15⋊3(C2×D4), (C5×D12)⋊6C2, (C3×D20)⋊6C2, (C4×D15)⋊7C2, C15⋊D4⋊3C2, (C6×D5)⋊2C22, (S3×C10)⋊2C22, C6.14(C22×D5), C10.14(C22×S3), (C2×S3×D5)⋊3C2, C2.17(C2×S3×D5), SmallGroup(240,138)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊D6
G = < a,b,c | a20=b6=c2=1, bab-1=a-1, cac=a9, cbc=b-1 >
Subgroups: 584 in 108 conjugacy classes, 34 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×D4, Dic5, C20, D10, D10, C2×C10, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C5×S3, C3×D5, D15, C30, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, S3×D4, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, D4×D5, C15⋊D4, C3×D20, C5×D12, C4×D15, C2×S3×D5, C20⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C22×D5, S3×D4, S3×D5, D4×D5, C2×S3×D5, C20⋊D6
►On 60 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)
(1 50 37 6 45 22)(2 49 38 5 46 21)(3 48 39 4 47 40)(7 44 23 20 51 36)(8 43 24 19 52 35)(9 42 25 18 53 34)(10 41 26 17 54 33)(11 60 27 16 55 32)(12 59 28 15 56 31)(13 58 29 14 57 30)
(1 27)(2 36)(3 25)(4 34)(5 23)(6 32)(7 21)(8 30)(9 39)(10 28)(11 37)(12 26)(13 35)(14 24)(15 33)(16 22)(17 31)(18 40)(19 29)(20 38)(41 59)(42 48)(43 57)(44 46)(45 55)(47 53)(49 51)(50 60)(52 58)(54 56)
G:=sub<Sym(60)| (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), (1,50,37,6,45,22)(2,49,38,5,46,21)(3,48,39,4,47,40)(7,44,23,20,51,36)(8,43,24,19,52,35)(9,42,25,18,53,34)(10,41,26,17,54,33)(11,60,27,16,55,32)(12,59,28,15,56,31)(13,58,29,14,57,30), (1,27)(2,36)(3,25)(4,34)(5,23)(6,32)(7,21)(8,30)(9,39)(10,28)(11,37)(12,26)(13,35)(14,24)(15,33)(16,22)(17,31)(18,40)(19,29)(20,38)(41,59)(42,48)(43,57)(44,46)(45,55)(47,53)(49,51)(50,60)(52,58)(54,56)>;
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), (1,50,37,6,45,22)(2,49,38,5,46,21)(3,48,39,4,47,40)(7,44,23,20,51,36)(8,43,24,19,52,35)(9,42,25,18,53,34)(10,41,26,17,54,33)(11,60,27,16,55,32)(12,59,28,15,56,31)(13,58,29,14,57,30), (1,27)(2,36)(3,25)(4,34)(5,23)(6,32)(7,21)(8,30)(9,39)(10,28)(11,37)(12,26)(13,35)(14,24)(15,33)(16,22)(17,31)(18,40)(19,29)(20,38)(41,59)(42,48)(43,57)(44,46)(45,55)(47,53)(49,51)(50,60)(52,58)(54,56) );
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)], [(1,50,37,6,45,22),(2,49,38,5,46,21),(3,48,39,4,47,40),(7,44,23,20,51,36),(8,43,24,19,52,35),(9,42,25,18,53,34),(10,41,26,17,54,33),(11,60,27,16,55,32),(12,59,28,15,56,31),(13,58,29,14,57,30)], [(1,27),(2,36),(3,25),(4,34),(5,23),(6,32),(7,21),(8,30),(9,39),(10,28),(11,37),(12,26),(13,35),(14,24),(15,33),(16,22),(17,31),(18,40),(19,29),(20,38),(41,59),(42,48),(43,57),(44,46),(45,55),(47,53),(49,51),(50,60),(52,58),(54,56)]])
C20⋊D6 is a maximal subgroup of
C40⋊14D6 C40⋊5D6 D24⋊6D5 C40⋊8D6 D15⋊D8 D30.8D4 D15⋊SD16 D60⋊C22 D20⋊24D6 D20⋊25D6 D20⋊26D6 S3×D4×D5 D20⋊13D6 D20⋊16D6 D20⋊17D6
C20⋊D6 is a maximal quotient of
C40⋊14D6 C40⋊5D6 D24⋊6D5 C40⋊8D6 Dic10.D6 Dic6.D10 D40⋊5S3 D30.3D4 D24⋊5D5 D30.4D4 D30.D4 Dic15⋊D4 Dic15.D4 D20⋊8Dic3 Dic15⋊8D4 D6⋊4Dic10 D30.2Q8 D30.7D4 Dic15⋊9D4 Dic15⋊2D4 D30⋊12D4 C60⋊10D4 Dic15.31D4 C12⋊2D20 C20⋊2D12 C20⋊Dic6 D30.27D4 D6⋊4D20 D30⋊4D4
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | 12 | 15A | 15B | 20A | 20B | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 6 | 6 | 10 | 10 | 15 | 15 | 2 | 2 | 30 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D10 | D10 | S3×D4 | S3×D5 | D4×D5 | C2×S3×D5 | C20⋊D6 |
| kernel | C20⋊D6 | C15⋊D4 | C3×D20 | C5×D12 | C4×D15 | C2×S3×D5 | D20 | D15 | D12 | C20 | D10 | C12 | D6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C20⋊D6 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 3 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 17 | 0 | 0 |
| 0 | 0 | 1 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 58 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 44 | 0 | 0 |
| 0 | 0 | 60 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,1,44,0,0,0,0,0,0,60,40,0,0,0,0,3,1],[1,60,0,0,0,0,1,0,0,0,0,0,0,0,44,1,0,0,0,0,17,17,0,0,0,0,0,0,1,0,0,0,0,0,58,60],[60,0,0,0,0,0,60,1,0,0,0,0,0,0,17,60,0,0,0,0,44,44,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;
C20⋊D6 in GAP, Magma, Sage, TeX
C_{20}\rtimes D_6 % in TeX
G:=Group("C20:D6"); // GroupNames label
G:=SmallGroup(240,138);
// by ID
G=gap.SmallGroup(240,138);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,218,116,50,490,6917]);
// Polycyclic
G:=Group<a,b,c|a^20=b^6=c^2=1,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=b^-1>;
// generators/relations