direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×C6×F5, C30⋊2C12, D5.C62, C5⋊(C6×C12), C10⋊(C3×C12), D5⋊(C3×C12), (C3×C30)⋊4C4, D10.(C3×C6), C15⋊3(C2×C12), (C3×D5)⋊3C12, (C6×D5).6C6, (C32×D5)⋊7C4, (C32×D5).7C22, (D5×C3×C6).5C2, (C3×C15)⋊10(C2×C4), (C3×D5).4(C2×C6), SmallGroup(360,145)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C3×C6×F5 |
Generators and relations for C3×C6×F5
G = < a,b,c,d | a3=b6=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 240 in 96 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C2×C4, C32, D5, C10, C12, C2×C6, C15, C3×C6, C3×C6, F5, D10, C2×C12, C3×D5, C30, C3×C12, C62, C2×F5, C3×C15, C3×F5, C6×D5, C6×C12, C32×D5, C3×C30, C6×F5, C32×F5, D5×C3×C6, C3×C6×F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, C3×C6, F5, C2×C12, C3×C12, C62, C2×F5, C3×F5, C6×C12, C6×F5, C32×F5, C3×C6×F5
►On 90 points
Generators in S90
(1 68 89)(2 69 90)(3 70 85)(4 71 86)(5 72 87)(6 67 88)(7 61 82)(8 62 83)(9 63 84)(10 64 79)(11 65 80)(12 66 81)(13 34 50)(14 35 51)(15 36 52)(16 31 53)(17 32 54)(18 33 49)(19 40 56)(20 41 57)(21 42 58)(22 37 59)(23 38 60)(24 39 55)(25 76 47)(26 77 48)(27 78 43)(28 73 44)(29 74 45)(30 75 46)
(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)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)
(1 25 37 79 36)(2 26 38 80 31)(3 27 39 81 32)(4 28 40 82 33)(5 29 41 83 34)(6 30 42 84 35)(7 49 71 73 56)(8 50 72 74 57)(9 51 67 75 58)(10 52 68 76 59)(11 53 69 77 60)(12 54 70 78 55)(13 87 45 20 62)(14 88 46 21 63)(15 89 47 22 64)(16 90 48 23 65)(17 85 43 24 66)(18 86 44 19 61)
(7 73 56 49)(8 74 57 50)(9 75 58 51)(10 76 59 52)(11 77 60 53)(12 78 55 54)(13 62 45 20)(14 63 46 21)(15 64 47 22)(16 65 48 23)(17 66 43 24)(18 61 44 19)(25 37 36 79)(26 38 31 80)(27 39 32 81)(28 40 33 82)(29 41 34 83)(30 42 35 84)
G:=sub<Sym(90)| (1,68,89)(2,69,90)(3,70,85)(4,71,86)(5,72,87)(6,67,88)(7,61,82)(8,62,83)(9,63,84)(10,64,79)(11,65,80)(12,66,81)(13,34,50)(14,35,51)(15,36,52)(16,31,53)(17,32,54)(18,33,49)(19,40,56)(20,41,57)(21,42,58)(22,37,59)(23,38,60)(24,39,55)(25,76,47)(26,77,48)(27,78,43)(28,73,44)(29,74,45)(30,75,46), (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)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90), (1,25,37,79,36)(2,26,38,80,31)(3,27,39,81,32)(4,28,40,82,33)(5,29,41,83,34)(6,30,42,84,35)(7,49,71,73,56)(8,50,72,74,57)(9,51,67,75,58)(10,52,68,76,59)(11,53,69,77,60)(12,54,70,78,55)(13,87,45,20,62)(14,88,46,21,63)(15,89,47,22,64)(16,90,48,23,65)(17,85,43,24,66)(18,86,44,19,61), (7,73,56,49)(8,74,57,50)(9,75,58,51)(10,76,59,52)(11,77,60,53)(12,78,55,54)(13,62,45,20)(14,63,46,21)(15,64,47,22)(16,65,48,23)(17,66,43,24)(18,61,44,19)(25,37,36,79)(26,38,31,80)(27,39,32,81)(28,40,33,82)(29,41,34,83)(30,42,35,84)>;
G:=Group( (1,68,89)(2,69,90)(3,70,85)(4,71,86)(5,72,87)(6,67,88)(7,61,82)(8,62,83)(9,63,84)(10,64,79)(11,65,80)(12,66,81)(13,34,50)(14,35,51)(15,36,52)(16,31,53)(17,32,54)(18,33,49)(19,40,56)(20,41,57)(21,42,58)(22,37,59)(23,38,60)(24,39,55)(25,76,47)(26,77,48)(27,78,43)(28,73,44)(29,74,45)(30,75,46), (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)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90), (1,25,37,79,36)(2,26,38,80,31)(3,27,39,81,32)(4,28,40,82,33)(5,29,41,83,34)(6,30,42,84,35)(7,49,71,73,56)(8,50,72,74,57)(9,51,67,75,58)(10,52,68,76,59)(11,53,69,77,60)(12,54,70,78,55)(13,87,45,20,62)(14,88,46,21,63)(15,89,47,22,64)(16,90,48,23,65)(17,85,43,24,66)(18,86,44,19,61), (7,73,56,49)(8,74,57,50)(9,75,58,51)(10,76,59,52)(11,77,60,53)(12,78,55,54)(13,62,45,20)(14,63,46,21)(15,64,47,22)(16,65,48,23)(17,66,43,24)(18,61,44,19)(25,37,36,79)(26,38,31,80)(27,39,32,81)(28,40,33,82)(29,41,34,83)(30,42,35,84) );
G=PermutationGroup([[(1,68,89),(2,69,90),(3,70,85),(4,71,86),(5,72,87),(6,67,88),(7,61,82),(8,62,83),(9,63,84),(10,64,79),(11,65,80),(12,66,81),(13,34,50),(14,35,51),(15,36,52),(16,31,53),(17,32,54),(18,33,49),(19,40,56),(20,41,57),(21,42,58),(22,37,59),(23,38,60),(24,39,55),(25,76,47),(26,77,48),(27,78,43),(28,73,44),(29,74,45),(30,75,46)], [(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),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90)], [(1,25,37,79,36),(2,26,38,80,31),(3,27,39,81,32),(4,28,40,82,33),(5,29,41,83,34),(6,30,42,84,35),(7,49,71,73,56),(8,50,72,74,57),(9,51,67,75,58),(10,52,68,76,59),(11,53,69,77,60),(12,54,70,78,55),(13,87,45,20,62),(14,88,46,21,63),(15,89,47,22,64),(16,90,48,23,65),(17,85,43,24,66),(18,86,44,19,61)], [(7,73,56,49),(8,74,57,50),(9,75,58,51),(10,76,59,52),(11,77,60,53),(12,78,55,54),(13,62,45,20),(14,63,46,21),(15,64,47,22),(16,65,48,23),(17,66,43,24),(18,61,44,19),(25,37,36,79),(26,38,31,80),(27,39,32,81),(28,40,33,82),(29,41,34,83),(30,42,35,84)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 5 | 6A | ··· | 6H | 6I | ··· | 6X | 10 | 12A | ··· | 12AF | 15A | ··· | 15H | 30A | ··· | 30H |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 5 | 5 | 1 | ··· | 1 | 5 | 5 | 5 | 5 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 5 | ··· | 5 | 4 | ··· | 4 | 4 | ··· | 4 |
90 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 | C3×C6×F5 | C32×F5 | D5×C3×C6 | C6×F5 | C32×D5 | C3×C30 | C3×F5 | C6×D5 | C3×D5 | C30 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 2 | 1 | 8 | 2 | 2 | 16 | 8 | 16 | 16 | 1 | 1 | 8 | 8 |
Matrix representation of C3×C6×F5 ►in GL5(𝔽61)
| 13 | 0 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 | 0 |
| 0 | 0 | 47 | 0 | 0 |
| 0 | 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 0 | 47 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 | 0 |
| 0 | 0 | 47 | 0 | 0 |
| 0 | 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 0 | 47 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 0 | 1 | 0 | 0 | 60 |
| 0 | 0 | 1 | 0 | 60 |
| 0 | 0 | 0 | 1 | 60 |
| 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(5,GF(61))| [13,0,0,0,0,0,47,0,0,0,0,0,47,0,0,0,0,0,47,0,0,0,0,0,47],[60,0,0,0,0,0,47,0,0,0,0,0,47,0,0,0,0,0,47,0,0,0,0,0,47],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,60,60,60,60],[50,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0] >;
C3×C6×F5 in GAP, Magma, Sage, TeX
C_3\times C_6\times F_5
% in TeX
G:=Group("C3xC6xF5"); // GroupNames label
G:=SmallGroup(360,145);
// by ID
G=gap.SmallGroup(360,145);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,-5,216,5189,317]);
// Polycyclic
G:=Group<a,b,c,d|a^3=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