direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C6×F5, D10⋊3C12, C10⋊(C2×C12), D5⋊(C2×C12), C5⋊(C22×C12), C30⋊3(C2×C4), (C2×C30)⋊4C4, (C6×D5)⋊7C4, (C2×C10)⋊4C12, C15⋊4(C22×C4), D5.(C22×C6), D10.7(C2×C6), (C3×D5).3C23, (C22×D5).4C6, (C6×D5).26C22, (D5×C2×C6).7C2, (C3×D5)⋊5(C2×C4), SmallGroup(240,200)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C2×C6×F5 |
Generators and relations for C2×C6×F5
G = < a,b,c,d | a2=b6=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 284 in 108 conjugacy classes, 64 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C23, D5, D5, C10, C12, C2×C6, C2×C6, C15, C22×C4, F5, D10, C2×C10, C2×C12, C22×C6, C3×D5, C3×D5, C30, C2×F5, C22×D5, C22×C12, C3×F5, C6×D5, C2×C30, C22×F5, C6×F5, D5×C2×C6, C2×C6×F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, F5, C2×C12, C22×C6, C2×F5, C22×C12, C3×F5, C22×F5, C6×F5, C2×C6×F5
►On 60 points
(1 59)(2 60)(3 55)(4 56)(5 57)(6 58)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 45)(14 46)(15 47)(16 48)(17 43)(18 44)(19 36)(20 31)(21 32)(22 33)(23 34)(24 35)(25 42)(26 37)(27 38)(28 39)(29 40)(30 41)
(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 43 37 49 36)(2 44 38 50 31)(3 45 39 51 32)(4 46 40 52 33)(5 47 41 53 34)(6 48 42 54 35)(7 19 59 17 26)(8 20 60 18 27)(9 21 55 13 28)(10 22 56 14 29)(11 23 57 15 30)(12 24 58 16 25)
(1 59)(2 60)(3 55)(4 56)(5 57)(6 58)(7 43 26 36)(8 44 27 31)(9 45 28 32)(10 46 29 33)(11 47 30 34)(12 48 25 35)(13 39 21 51)(14 40 22 52)(15 41 23 53)(16 42 24 54)(17 37 19 49)(18 38 20 50)
G:=sub<Sym(60)| (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,45)(14,46)(15,47)(16,48)(17,43)(18,44)(19,36)(20,31)(21,32)(22,33)(23,34)(24,35)(25,42)(26,37)(27,38)(28,39)(29,40)(30,41), (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), (1,43,37,49,36)(2,44,38,50,31)(3,45,39,51,32)(4,46,40,52,33)(5,47,41,53,34)(6,48,42,54,35)(7,19,59,17,26)(8,20,60,18,27)(9,21,55,13,28)(10,22,56,14,29)(11,23,57,15,30)(12,24,58,16,25), (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,43,26,36)(8,44,27,31)(9,45,28,32)(10,46,29,33)(11,47,30,34)(12,48,25,35)(13,39,21,51)(14,40,22,52)(15,41,23,53)(16,42,24,54)(17,37,19,49)(18,38,20,50)>;
G:=Group( (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,45)(14,46)(15,47)(16,48)(17,43)(18,44)(19,36)(20,31)(21,32)(22,33)(23,34)(24,35)(25,42)(26,37)(27,38)(28,39)(29,40)(30,41), (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), (1,43,37,49,36)(2,44,38,50,31)(3,45,39,51,32)(4,46,40,52,33)(5,47,41,53,34)(6,48,42,54,35)(7,19,59,17,26)(8,20,60,18,27)(9,21,55,13,28)(10,22,56,14,29)(11,23,57,15,30)(12,24,58,16,25), (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,43,26,36)(8,44,27,31)(9,45,28,32)(10,46,29,33)(11,47,30,34)(12,48,25,35)(13,39,21,51)(14,40,22,52)(15,41,23,53)(16,42,24,54)(17,37,19,49)(18,38,20,50) );
G=PermutationGroup([[(1,59),(2,60),(3,55),(4,56),(5,57),(6,58),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,45),(14,46),(15,47),(16,48),(17,43),(18,44),(19,36),(20,31),(21,32),(22,33),(23,34),(24,35),(25,42),(26,37),(27,38),(28,39),(29,40),(30,41)], [(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)], [(1,43,37,49,36),(2,44,38,50,31),(3,45,39,51,32),(4,46,40,52,33),(5,47,41,53,34),(6,48,42,54,35),(7,19,59,17,26),(8,20,60,18,27),(9,21,55,13,28),(10,22,56,14,29),(11,23,57,15,30),(12,24,58,16,25)], [(1,59),(2,60),(3,55),(4,56),(5,57),(6,58),(7,43,26,36),(8,44,27,31),(9,45,28,32),(10,46,29,33),(11,47,30,34),(12,48,25,35),(13,39,21,51),(14,40,22,52),(15,41,23,53),(16,42,24,54),(17,37,19,49),(18,38,20,50)]])
C2×C6×F5 is a maximal subgroup of
D10.20D12
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | ··· | 4H | 5 | 6A | ··· | 6F | 6G | ··· | 6N | 10A | 10B | 10C | 12A | ··· | 12P | 15A | 15B | 30A | ··· | 30F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 5 | ··· | 5 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 4 | 4 | 5 | ··· | 5 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | F5 | C2×F5 | C3×F5 | C6×F5 |
| kernel | C2×C6×F5 | C6×F5 | D5×C2×C6 | C22×F5 | C6×D5 | C2×C30 | C2×F5 | C22×D5 | D10 | C2×C10 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 6 | 1 | 2 | 6 | 2 | 12 | 2 | 12 | 4 | 1 | 3 | 2 | 6 |
Matrix representation of C2×C6×F5 ►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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 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 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 60 | 60 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 1 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[13,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,1,0,0,0,60,0,0,1,0,0,60,0,0,0],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,60,0,0,1,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,60,0,1] >;
C2×C6×F5 in GAP, Magma, Sage, TeX
C_2\times C_6\times F_5
% in TeX
G:=Group("C2xC6xF5"); // GroupNames label
G:=SmallGroup(240,200);
// by ID
G=gap.SmallGroup(240,200);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-5,144,3461,317]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations