direct product, metabelian, supersoluble, monomial, A-group
Aliases: C5×C6.D6, C30.36D6, C3⋊S3⋊1C20, C3⋊1(S3×C20), C10.14S32, C15⋊12(C4×S3), C6.2(S3×C10), C32⋊3(C2×C20), (C5×Dic3)⋊5S3, Dic3⋊2(C5×S3), (C3×Dic3)⋊3C10, (Dic3×C15)⋊9C2, (C3×C30).28C22, C2.2(C5×S32), (C5×C3⋊S3)⋊6C4, (C3×C15)⋊25(C2×C4), (C2×C3⋊S3).1C10, (C10×C3⋊S3).3C2, (C3×C6).2(C2×C10), SmallGroup(360,73)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C5×C6.D6 |
Generators and relations for C5×C6.D6
G = < a,b,c,d | a5=b6=d2=1, c6=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c5 >
Subgroups: 220 in 74 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, C2×C4, C32, C10, C10, Dic3, C12, D6, C15, C15, C3⋊S3, C3×C6, C20, C2×C10, C4×S3, C5×S3, C30, C30, C3×Dic3, C2×C3⋊S3, C2×C20, C3×C15, C5×Dic3, C60, S3×C10, C6.D6, C5×C3⋊S3, C3×C30, S3×C20, Dic3×C15, C10×C3⋊S3, C5×C6.D6
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C10, D6, C20, C2×C10, C4×S3, C5×S3, S32, C2×C20, S3×C10, C6.D6, S3×C20, C5×S32, C5×C6.D6
►On 60 points
(1 44 26 20 50)(2 45 27 21 51)(3 46 28 22 52)(4 47 29 23 53)(5 48 30 24 54)(6 37 31 13 55)(7 38 32 14 56)(8 39 33 15 57)(9 40 34 16 58)(10 41 35 17 59)(11 42 36 18 60)(12 43 25 19 49)
(1 11 9 7 5 3)(2 4 6 8 10 12)(13 15 17 19 21 23)(14 24 22 20 18 16)(25 27 29 31 33 35)(26 36 34 32 30 28)(37 39 41 43 45 47)(38 48 46 44 42 40)(49 51 53 55 57 59)(50 60 58 56 54 52)
(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 9)(3 7)(4 12)(6 10)(13 17)(14 22)(16 20)(19 23)(25 29)(26 34)(28 32)(31 35)(37 41)(38 46)(40 44)(43 47)(49 53)(50 58)(52 56)(55 59)
G:=sub<Sym(60)| (1,44,26,20,50)(2,45,27,21,51)(3,46,28,22,52)(4,47,29,23,53)(5,48,30,24,54)(6,37,31,13,55)(7,38,32,14,56)(8,39,33,15,57)(9,40,34,16,58)(10,41,35,17,59)(11,42,36,18,60)(12,43,25,19,49), (1,11,9,7,5,3)(2,4,6,8,10,12)(13,15,17,19,21,23)(14,24,22,20,18,16)(25,27,29,31,33,35)(26,36,34,32,30,28)(37,39,41,43,45,47)(38,48,46,44,42,40)(49,51,53,55,57,59)(50,60,58,56,54,52), (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), (1,9)(3,7)(4,12)(6,10)(13,17)(14,22)(16,20)(19,23)(25,29)(26,34)(28,32)(31,35)(37,41)(38,46)(40,44)(43,47)(49,53)(50,58)(52,56)(55,59)>;
G:=Group( (1,44,26,20,50)(2,45,27,21,51)(3,46,28,22,52)(4,47,29,23,53)(5,48,30,24,54)(6,37,31,13,55)(7,38,32,14,56)(8,39,33,15,57)(9,40,34,16,58)(10,41,35,17,59)(11,42,36,18,60)(12,43,25,19,49), (1,11,9,7,5,3)(2,4,6,8,10,12)(13,15,17,19,21,23)(14,24,22,20,18,16)(25,27,29,31,33,35)(26,36,34,32,30,28)(37,39,41,43,45,47)(38,48,46,44,42,40)(49,51,53,55,57,59)(50,60,58,56,54,52), (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), (1,9)(3,7)(4,12)(6,10)(13,17)(14,22)(16,20)(19,23)(25,29)(26,34)(28,32)(31,35)(37,41)(38,46)(40,44)(43,47)(49,53)(50,58)(52,56)(55,59) );
G=PermutationGroup([[(1,44,26,20,50),(2,45,27,21,51),(3,46,28,22,52),(4,47,29,23,53),(5,48,30,24,54),(6,37,31,13,55),(7,38,32,14,56),(8,39,33,15,57),(9,40,34,16,58),(10,41,35,17,59),(11,42,36,18,60),(12,43,25,19,49)], [(1,11,9,7,5,3),(2,4,6,8,10,12),(13,15,17,19,21,23),(14,24,22,20,18,16),(25,27,29,31,33,35),(26,36,34,32,30,28),(37,39,41,43,45,47),(38,48,46,44,42,40),(49,51,53,55,57,59),(50,60,58,56,54,52)], [(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)], [(1,9),(3,7),(4,12),(6,10),(13,17),(14,22),(16,20),(19,23),(25,29),(26,34),(28,32),(31,35),(37,41),(38,46),(40,44),(43,47),(49,53),(50,58),(52,56),(55,59)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | 12B | 12C | 12D | 15A | ··· | 15H | 15I | 15J | 15K | 15L | 20A | ··· | 20P | 30A | ··· | 30H | 30I | 30J | 30K | 30L | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 4 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 9 | ··· | 9 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 3 | ··· | 3 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | S3 | D6 | C4×S3 | C5×S3 | S3×C10 | S3×C20 | S32 | C6.D6 | C5×S32 | C5×C6.D6 |
| kernel | C5×C6.D6 | Dic3×C15 | C10×C3⋊S3 | C5×C3⋊S3 | C6.D6 | C3×Dic3 | C2×C3⋊S3 | C3⋊S3 | C5×Dic3 | C30 | C15 | Dic3 | C6 | C3 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 | 2 | 2 | 4 | 8 | 8 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C5×C6.D6 ►in GL4(𝔽61) generated by
| 58 | 0 | 0 | 0 |
| 0 | 58 | 0 | 0 |
| 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 20 |
| 0 | 1 | 0 | 0 |
| 60 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 50 | 11 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 60 |
| 1 | 60 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 60 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,20,0,0,0,0,20],[0,60,0,0,1,1,0,0,0,0,1,0,0,0,0,1],[50,0,0,0,11,11,0,0,0,0,0,60,0,0,1,60],[1,0,0,0,60,60,0,0,0,0,60,0,0,0,60,1] >;
C5×C6.D6 in GAP, Magma, Sage, TeX
C_5\times C_6.D_6
% in TeX
G:=Group("C5xC6.D6"); // GroupNames label
G:=SmallGroup(360,73);
// by ID
G=gap.SmallGroup(360,73);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-3,-3,120,127,1210,8645]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^6=d^2=1,c^6=b^3,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