non-abelian, soluble, monomial, A-group
Aliases: C52⋊2Dic3, (C5×C10).S3, C52⋊C3⋊3C4, C2.(C52⋊S3), (C2×C52⋊C3).1C2, SmallGroup(300,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C52⋊C3 — C52⋊2Dic3 |
Generators and relations for C52⋊2Dic3
G = < a,b,c,d | a5=b5=c6=1, d2=c3, cbc-1=ab=ba, cac-1=a3b2, ad=da, dbd-1=a-1b-1, dcd-1=c-1 >
Character table of C52⋊2Dic3
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 5F | 6 | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | |
| size | 1 | 1 | 50 | 15 | 15 | 3 | 3 | 3 | 3 | 6 | 6 | 50 | 3 | 3 | 3 | 3 | 6 | 6 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
| ρ4 | 1 | -1 | 1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | i | i | i | i | -i | -i | -i | -i | linear of order 4 |
| ρ5 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ7 | 3 | 3 | 0 | 1 | 1 | 2ζ54+ζ52 | ζ53+2ζ5 | 2ζ52+ζ5 | ζ54+2ζ53 | 1+√5/2 | 1-√5/2 | 0 | 2ζ54+ζ52 | ζ53+2ζ5 | ζ54+2ζ53 | 2ζ52+ζ5 | 1+√5/2 | 1-√5/2 | ζ52 | ζ53 | ζ54 | ζ5 | ζ5 | ζ52 | ζ53 | ζ54 | complex lifted from C52⋊S3 |
| ρ8 | 3 | 3 | 0 | -1 | -1 | 2ζ52+ζ5 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ54+ζ52 | 1-√5/2 | 1+√5/2 | 0 | 2ζ52+ζ5 | ζ54+2ζ53 | 2ζ54+ζ52 | ζ53+2ζ5 | 1-√5/2 | 1+√5/2 | -ζ5 | -ζ54 | -ζ52 | -ζ53 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | complex lifted from C52⋊S3 |
| ρ9 | 3 | 3 | 0 | 1 | 1 | ζ53+2ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | 2ζ52+ζ5 | 1+√5/2 | 1-√5/2 | 0 | ζ53+2ζ5 | 2ζ54+ζ52 | 2ζ52+ζ5 | ζ54+2ζ53 | 1+√5/2 | 1-√5/2 | ζ53 | ζ52 | ζ5 | ζ54 | ζ54 | ζ53 | ζ52 | ζ5 | complex lifted from C52⋊S3 |
| ρ10 | 3 | 3 | 0 | -1 | -1 | ζ53+2ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | 2ζ52+ζ5 | 1+√5/2 | 1-√5/2 | 0 | ζ53+2ζ5 | 2ζ54+ζ52 | 2ζ52+ζ5 | ζ54+2ζ53 | 1+√5/2 | 1-√5/2 | -ζ53 | -ζ52 | -ζ5 | -ζ54 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | complex lifted from C52⋊S3 |
| ρ11 | 3 | 3 | 0 | -1 | -1 | 2ζ54+ζ52 | ζ53+2ζ5 | 2ζ52+ζ5 | ζ54+2ζ53 | 1+√5/2 | 1-√5/2 | 0 | 2ζ54+ζ52 | ζ53+2ζ5 | ζ54+2ζ53 | 2ζ52+ζ5 | 1+√5/2 | 1-√5/2 | -ζ52 | -ζ53 | -ζ54 | -ζ5 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | complex lifted from C52⋊S3 |
| ρ12 | 3 | 3 | 0 | 1 | 1 | ζ54+2ζ53 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ53+2ζ5 | 1-√5/2 | 1+√5/2 | 0 | ζ54+2ζ53 | 2ζ52+ζ5 | ζ53+2ζ5 | 2ζ54+ζ52 | 1-√5/2 | 1+√5/2 | ζ54 | ζ5 | ζ53 | ζ52 | ζ52 | ζ54 | ζ5 | ζ53 | complex lifted from C52⋊S3 |
| ρ13 | 3 | 3 | 0 | 1 | 1 | 2ζ52+ζ5 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ54+ζ52 | 1-√5/2 | 1+√5/2 | 0 | 2ζ52+ζ5 | ζ54+2ζ53 | 2ζ54+ζ52 | ζ53+2ζ5 | 1-√5/2 | 1+√5/2 | ζ5 | ζ54 | ζ52 | ζ53 | ζ53 | ζ5 | ζ54 | ζ52 | complex lifted from C52⋊S3 |
| ρ14 | 3 | 3 | 0 | -1 | -1 | ζ54+2ζ53 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ53+2ζ5 | 1-√5/2 | 1+√5/2 | 0 | ζ54+2ζ53 | 2ζ52+ζ5 | ζ53+2ζ5 | 2ζ54+ζ52 | 1-√5/2 | 1+√5/2 | -ζ54 | -ζ5 | -ζ53 | -ζ52 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | complex lifted from C52⋊S3 |
| ρ15 | 3 | -3 | 0 | i | -i | 2ζ52+ζ5 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ54+ζ52 | 1-√5/2 | 1+√5/2 | 0 | -2ζ52-ζ5 | -ζ54-2ζ53 | -2ζ54-ζ52 | -ζ53-2ζ5 | -1+√5/2 | -1-√5/2 | ζ43ζ5 | ζ43ζ54 | ζ43ζ52 | ζ43ζ53 | ζ4ζ53 | ζ4ζ5 | ζ4ζ54 | ζ4ζ52 | complex faithful |
| ρ16 | 3 | -3 | 0 | -i | i | ζ54+2ζ53 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ53+2ζ5 | 1-√5/2 | 1+√5/2 | 0 | -ζ54-2ζ53 | -2ζ52-ζ5 | -ζ53-2ζ5 | -2ζ54-ζ52 | -1+√5/2 | -1-√5/2 | ζ4ζ54 | ζ4ζ5 | ζ4ζ53 | ζ4ζ52 | ζ43ζ52 | ζ43ζ54 | ζ43ζ5 | ζ43ζ53 | complex faithful |
| ρ17 | 3 | -3 | 0 | -i | i | 2ζ54+ζ52 | ζ53+2ζ5 | 2ζ52+ζ5 | ζ54+2ζ53 | 1+√5/2 | 1-√5/2 | 0 | -2ζ54-ζ52 | -ζ53-2ζ5 | -ζ54-2ζ53 | -2ζ52-ζ5 | -1-√5/2 | -1+√5/2 | ζ4ζ52 | ζ4ζ53 | ζ4ζ54 | ζ4ζ5 | ζ43ζ5 | ζ43ζ52 | ζ43ζ53 | ζ43ζ54 | complex faithful |
| ρ18 | 3 | -3 | 0 | -i | i | ζ53+2ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | 2ζ52+ζ5 | 1+√5/2 | 1-√5/2 | 0 | -ζ53-2ζ5 | -2ζ54-ζ52 | -2ζ52-ζ5 | -ζ54-2ζ53 | -1-√5/2 | -1+√5/2 | ζ4ζ53 | ζ4ζ52 | ζ4ζ5 | ζ4ζ54 | ζ43ζ54 | ζ43ζ53 | ζ43ζ52 | ζ43ζ5 | complex faithful |
| ρ19 | 3 | -3 | 0 | i | -i | 2ζ54+ζ52 | ζ53+2ζ5 | 2ζ52+ζ5 | ζ54+2ζ53 | 1+√5/2 | 1-√5/2 | 0 | -2ζ54-ζ52 | -ζ53-2ζ5 | -ζ54-2ζ53 | -2ζ52-ζ5 | -1-√5/2 | -1+√5/2 | ζ43ζ52 | ζ43ζ53 | ζ43ζ54 | ζ43ζ5 | ζ4ζ5 | ζ4ζ52 | ζ4ζ53 | ζ4ζ54 | complex faithful |
| ρ20 | 3 | -3 | 0 | -i | i | 2ζ52+ζ5 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ54+ζ52 | 1-√5/2 | 1+√5/2 | 0 | -2ζ52-ζ5 | -ζ54-2ζ53 | -2ζ54-ζ52 | -ζ53-2ζ5 | -1+√5/2 | -1-√5/2 | ζ4ζ5 | ζ4ζ54 | ζ4ζ52 | ζ4ζ53 | ζ43ζ53 | ζ43ζ5 | ζ43ζ54 | ζ43ζ52 | complex faithful |
| ρ21 | 3 | -3 | 0 | i | -i | ζ54+2ζ53 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ53+2ζ5 | 1-√5/2 | 1+√5/2 | 0 | -ζ54-2ζ53 | -2ζ52-ζ5 | -ζ53-2ζ5 | -2ζ54-ζ52 | -1+√5/2 | -1-√5/2 | ζ43ζ54 | ζ43ζ5 | ζ43ζ53 | ζ43ζ52 | ζ4ζ52 | ζ4ζ54 | ζ4ζ5 | ζ4ζ53 | complex faithful |
| ρ22 | 3 | -3 | 0 | i | -i | ζ53+2ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | 2ζ52+ζ5 | 1+√5/2 | 1-√5/2 | 0 | -ζ53-2ζ5 | -2ζ54-ζ52 | -2ζ52-ζ5 | -ζ54-2ζ53 | -1-√5/2 | -1+√5/2 | ζ43ζ53 | ζ43ζ52 | ζ43ζ5 | ζ43ζ54 | ζ4ζ54 | ζ4ζ53 | ζ4ζ52 | ζ4ζ5 | complex faithful |
| ρ23 | 6 | 6 | 0 | 0 | 0 | 1+√5 | 1+√5 | 1-√5 | 1-√5 | -3-√5/2 | -3+√5/2 | 0 | 1+√5 | 1+√5 | 1-√5 | 1-√5 | -3-√5/2 | -3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊S3 |
| ρ24 | 6 | 6 | 0 | 0 | 0 | 1-√5 | 1-√5 | 1+√5 | 1+√5 | -3+√5/2 | -3-√5/2 | 0 | 1-√5 | 1-√5 | 1+√5 | 1+√5 | -3+√5/2 | -3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊S3 |
| ρ25 | 6 | -6 | 0 | 0 | 0 | 1-√5 | 1-√5 | 1+√5 | 1+√5 | -3+√5/2 | -3-√5/2 | 0 | -1+√5 | -1+√5 | -1-√5 | -1-√5 | 3-√5/2 | 3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ26 | 6 | -6 | 0 | 0 | 0 | 1+√5 | 1+√5 | 1-√5 | 1-√5 | -3-√5/2 | -3+√5/2 | 0 | -1-√5 | -1-√5 | -1+√5 | -1+√5 | 3+√5/2 | 3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 60 points
Generators in S60
(2 42 48 35 54)(3 49 36 43 37)(5 39 45 32 51)(6 52 33 46 40)(7 55 19 13 26)(9 28 15 21 57)(10 58 22 16 29)(12 25 18 24 60)
(1 34 41 53 47)(2 35 42 54 48)(3 49 36 43 37)(4 31 38 50 44)(5 32 39 51 45)(6 52 33 46 40)(7 55 19 13 26)(8 20 27 56 14)(9 21 28 57 15)(10 58 22 16 29)(11 23 30 59 17)(12 24 25 60 18)
(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 59 4 56)(2 58 5 55)(3 57 6 60)(7 54 10 51)(8 53 11 50)(9 52 12 49)(13 48 16 45)(14 47 17 44)(15 46 18 43)(19 42 22 39)(20 41 23 38)(21 40 24 37)(25 36 28 33)(26 35 29 32)(27 34 30 31)
G:=sub<Sym(60)| (2,42,48,35,54)(3,49,36,43,37)(5,39,45,32,51)(6,52,33,46,40)(7,55,19,13,26)(9,28,15,21,57)(10,58,22,16,29)(12,25,18,24,60), (1,34,41,53,47)(2,35,42,54,48)(3,49,36,43,37)(4,31,38,50,44)(5,32,39,51,45)(6,52,33,46,40)(7,55,19,13,26)(8,20,27,56,14)(9,21,28,57,15)(10,58,22,16,29)(11,23,30,59,17)(12,24,25,60,18), (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), (1,59,4,56)(2,58,5,55)(3,57,6,60)(7,54,10,51)(8,53,11,50)(9,52,12,49)(13,48,16,45)(14,47,17,44)(15,46,18,43)(19,42,22,39)(20,41,23,38)(21,40,24,37)(25,36,28,33)(26,35,29,32)(27,34,30,31)>;
G:=Group( (2,42,48,35,54)(3,49,36,43,37)(5,39,45,32,51)(6,52,33,46,40)(7,55,19,13,26)(9,28,15,21,57)(10,58,22,16,29)(12,25,18,24,60), (1,34,41,53,47)(2,35,42,54,48)(3,49,36,43,37)(4,31,38,50,44)(5,32,39,51,45)(6,52,33,46,40)(7,55,19,13,26)(8,20,27,56,14)(9,21,28,57,15)(10,58,22,16,29)(11,23,30,59,17)(12,24,25,60,18), (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), (1,59,4,56)(2,58,5,55)(3,57,6,60)(7,54,10,51)(8,53,11,50)(9,52,12,49)(13,48,16,45)(14,47,17,44)(15,46,18,43)(19,42,22,39)(20,41,23,38)(21,40,24,37)(25,36,28,33)(26,35,29,32)(27,34,30,31) );
G=PermutationGroup([[(2,42,48,35,54),(3,49,36,43,37),(5,39,45,32,51),(6,52,33,46,40),(7,55,19,13,26),(9,28,15,21,57),(10,58,22,16,29),(12,25,18,24,60)], [(1,34,41,53,47),(2,35,42,54,48),(3,49,36,43,37),(4,31,38,50,44),(5,32,39,51,45),(6,52,33,46,40),(7,55,19,13,26),(8,20,27,56,14),(9,21,28,57,15),(10,58,22,16,29),(11,23,30,59,17),(12,24,25,60,18)], [(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)], [(1,59,4,56),(2,58,5,55),(3,57,6,60),(7,54,10,51),(8,53,11,50),(9,52,12,49),(13,48,16,45),(14,47,17,44),(15,46,18,43),(19,42,22,39),(20,41,23,38),(21,40,24,37),(25,36,28,33),(26,35,29,32),(27,34,30,31)]])
Matrix representation of C52⋊2Dic3 ►in GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 34 | 0 | 0 |
| 0 | 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 0 | 20 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 34 |
| 1 | 60 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 49 | 53 | 0 | 0 | 0 |
| 41 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,34,0,0,0,0,0,34,0,0,0,0,0,20],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,34],[1,1,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,60,0,0,1,0,0],[49,41,0,0,0,53,12,0,0,0,0,0,0,60,0,0,0,60,0,0,0,0,0,0,1] >;
C52⋊2Dic3 in GAP, Magma, Sage, TeX
C_5^2\rtimes_2{\rm Dic}_3 % in TeX
G:=Group("C5^2:2Dic3"); // GroupNames label
G:=SmallGroup(300,13);
// by ID
G=gap.SmallGroup(300,13);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,5,10,122,973,7204,1439]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^6=1,d^2=c^3,c*b*c^-1=a*b=b*a,c*a*c^-1=a^3*b^2,a*d=d*a,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of C52⋊2Dic3 in TeX
Character table of C52⋊2Dic3 in TeX