direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D4×D5, D20⋊3C6, C12⋊6D10, C60⋊6C22, C30.42C23, C20⋊(C2×C6), C5⋊2(C6×D4), C4⋊1(C6×D5), (C5×D4)⋊2C6, (C4×D5)⋊1C6, (C2×C6)⋊4D10, C15⋊14(C2×D4), C5⋊D4⋊1C6, (D5×C12)⋊6C2, (D4×C15)⋊5C2, D10⋊2(C2×C6), (C3×D20)⋊9C2, C22⋊2(C6×D5), (C2×C30)⋊6C22, Dic5⋊1(C2×C6), (C22×D5)⋊3C6, (C6×D5)⋊10C22, C10.5(C22×C6), C6.42(C22×D5), (C3×Dic5)⋊8C22, (D5×C2×C6)⋊6C2, C2.6(D5×C2×C6), (C2×C10)⋊2(C2×C6), (C3×C5⋊D4)⋊5C2, SmallGroup(240,159)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4×D5
G = < a,b,c,d,e | a3=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 340 in 108 conjugacy classes, 50 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, D4, D4, C23, D5, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×D4, Dic5, C20, D10, D10, D10, C2×C10, C2×C12, C3×D4, C3×D4, C22×C6, C3×D5, C3×D5, C30, C30, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, C6×D4, C3×Dic5, C60, C6×D5, C6×D5, C6×D5, C2×C30, D4×D5, D5×C12, C3×D20, C3×C5⋊D4, D4×C15, D5×C2×C6, C3×D4×D5
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, D10, C3×D4, C22×C6, C3×D5, C22×D5, C6×D4, C6×D5, D4×D5, D5×C2×C6, C3×D4×D5
►On 60 points
(1 24 14)(2 25 15)(3 21 11)(4 22 12)(5 23 13)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)
(1 34 9 39)(2 35 10 40)(3 31 6 36)(4 32 7 37)(5 33 8 38)(11 41 16 46)(12 42 17 47)(13 43 18 48)(14 44 19 49)(15 45 20 50)(21 51 26 56)(22 52 27 57)(23 53 28 58)(24 54 29 59)(25 55 30 60)
(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)
G:=sub<Sym(60)| (1,24,14)(2,25,15)(3,21,11)(4,22,12)(5,23,13)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (1,34,9,39)(2,35,10,40)(3,31,6,36)(4,32,7,37)(5,33,8,38)(11,41,16,46)(12,42,17,47)(13,43,18,48)(14,44,19,49)(15,45,20,50)(21,51,26,56)(22,52,27,57)(23,53,28,58)(24,54,29,59)(25,55,30,60), (31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)>;
G:=Group( (1,24,14)(2,25,15)(3,21,11)(4,22,12)(5,23,13)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (1,34,9,39)(2,35,10,40)(3,31,6,36)(4,32,7,37)(5,33,8,38)(11,41,16,46)(12,42,17,47)(13,43,18,48)(14,44,19,49)(15,45,20,50)(21,51,26,56)(22,52,27,57)(23,53,28,58)(24,54,29,59)(25,55,30,60), (31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59) );
G=PermutationGroup([[(1,24,14),(2,25,15),(3,21,11),(4,22,12),(5,23,13),(6,26,16),(7,27,17),(8,28,18),(9,29,19),(10,30,20),(31,51,41),(32,52,42),(33,53,43),(34,54,44),(35,55,45),(36,56,46),(37,57,47),(38,58,48),(39,59,49),(40,60,50)], [(1,34,9,39),(2,35,10,40),(3,31,6,36),(4,32,7,37),(5,33,8,38),(11,41,16,46),(12,42,17,47),(13,43,18,48),(14,44,19,49),(15,45,20,50),(21,51,26,56),(22,52,27,57),(23,53,28,58),(24,54,29,59),(25,55,30,60)], [(31,36),(32,37),(33,38),(34,39),(35,40),(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60)], [(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)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29),(32,35),(33,34),(37,40),(38,39),(42,45),(43,44),(47,50),(48,49),(52,55),(53,54),(57,60),(58,59)]])
C3×D4×D5 is a maximal subgroup of
D20⋊Dic3 D12⋊10D10 D20.9D6 D20⋊13D6 D20⋊14D6
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | 20B | 30A | 30B | 30C | 30D | 30E | ··· | 30L | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 2 | 2 | 5 | 5 | 10 | 10 | 1 | 1 | 2 | 10 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D5 | D10 | D10 | C3×D4 | C3×D5 | C6×D5 | C6×D5 | D4×D5 | C3×D4×D5 |
| kernel | C3×D4×D5 | D5×C12 | C3×D20 | C3×C5⋊D4 | D4×C15 | D5×C2×C6 | D4×D5 | C4×D5 | D20 | C5⋊D4 | C5×D4 | C22×D5 | C3×D5 | C3×D4 | C12 | C2×C6 | D5 | D4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 2 | 4 |
Matrix representation of C3×D4×D5 ►in GL4(𝔽61) generated by
| 47 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 |
| 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 47 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 20 |
| 0 | 0 | 6 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 41 |
| 0 | 0 | 0 | 60 |
| 0 | 1 | 0 | 0 |
| 60 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,47,0,0,0,0,47],[60,0,0,0,0,60,0,0,0,0,60,6,0,0,20,1],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,41,60],[0,60,0,0,1,17,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;
C3×D4×D5 in GAP, Magma, Sage, TeX
C_3\times D_4\times D_5
% in TeX
G:=Group("C3xD4xD5"); // GroupNames label
G:=SmallGroup(240,159);
// by ID
G=gap.SmallGroup(240,159);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-5,260,6917]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations