direct product, metabelian, soluble, monomial, A-group
Aliases: C4×D5×A4, C20⋊2(C2×A4), C22⋊(D5×C12), (A4×C20)⋊5C2, (C22×C20)⋊2C6, Dic5⋊2(C2×A4), (A4×Dic5)⋊5C2, D10.5(C2×A4), (C2×A4).14D10, (C22×D5)⋊4C12, C10.2(C22×A4), (C23×D5).2C6, C23.11(C6×D5), (C22×Dic5)⋊2C6, (C10×A4).14C22, C5⋊2(C2×C4×A4), (D5×C22×C4)⋊C3, C2.1(C2×D5×A4), (C2×D5×A4).4C2, (C5×A4)⋊8(C2×C4), (C2×C10)⋊3(C2×C12), (C22×C4)⋊2(C3×D5), (C22×C10).2(C2×C6), SmallGroup(480,1036)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C10 — C4×D5×A4 |
Generators and relations for C4×D5×A4
G = < a,b,c,d,e,f | a4=b5=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >
Subgroups: 780 in 132 conjugacy classes, 33 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C2×C4, C23, C23, D5, D5, C10, C10, C12, A4, C2×C6, C15, C22×C4, C22×C4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C12, C2×A4, C2×A4, C3×D5, C30, C23×C4, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×D5, C22×C10, C4×A4, C4×A4, C22×A4, C3×Dic5, C60, C5×A4, C6×D5, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, C2×C4×A4, D5×C12, D5×A4, C10×A4, D5×C22×C4, A4×Dic5, A4×C20, C2×D5×A4, C4×D5×A4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D5, C12, A4, C2×C6, D10, C2×C12, C2×A4, C3×D5, C4×D5, C4×A4, C22×A4, C6×D5, C2×C4×A4, D5×C12, D5×A4, C2×D5×A4, C4×D5×A4
►On 60 points
Generators in S60
(1 39 9 34)(2 40 10 35)(3 36 6 31)(4 37 7 32)(5 38 8 33)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 56 26 51)(22 57 27 52)(23 58 28 53)(24 59 29 54)(25 60 30 55)
(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 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)
(1 9)(2 10)(3 6)(4 7)(5 8)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(51 56)(52 57)(53 58)(54 59)(55 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)
G:=sub<Sym(60)| (1,39,9,34)(2,40,10,35)(3,36,6,31)(4,37,7,32)(5,38,8,33)(11,46,16,41)(12,47,17,42)(13,48,18,43)(14,49,19,44)(15,50,20,45)(21,56,26,51)(22,57,27,52)(23,58,28,53)(24,59,29,54)(25,60,30,55), (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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50), (1,9)(2,10)(3,6)(4,7)(5,8)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(51,56)(52,57)(53,58)(54,59)(55,60), (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)>;
G:=Group( (1,39,9,34)(2,40,10,35)(3,36,6,31)(4,37,7,32)(5,38,8,33)(11,46,16,41)(12,47,17,42)(13,48,18,43)(14,49,19,44)(15,50,20,45)(21,56,26,51)(22,57,27,52)(23,58,28,53)(24,59,29,54)(25,60,30,55), (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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50), (1,9)(2,10)(3,6)(4,7)(5,8)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(51,56)(52,57)(53,58)(54,59)(55,60), (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) );
G=PermutationGroup([[(1,39,9,34),(2,40,10,35),(3,36,6,31),(4,37,7,32),(5,38,8,33),(11,46,16,41),(12,47,17,42),(13,48,18,43),(14,49,19,44),(15,50,20,45),(21,56,26,51),(22,57,27,52),(23,58,28,53),(24,59,29,54),(25,60,30,55)], [(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)], [(1,8),(2,7),(3,6),(4,10),(5,9),(11,16),(12,20),(13,19),(14,18),(15,17),(21,26),(22,30),(23,29),(24,28),(25,27),(31,36),(32,40),(33,39),(34,38),(35,37),(41,46),(42,50),(43,49),(44,48),(45,47),(51,56),(52,60),(53,59),(54,58),(55,57)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,16),(12,17),(13,18),(14,19),(15,20),(31,36),(32,37),(33,38),(34,39),(35,40),(41,46),(42,47),(43,48),(44,49),(45,50)], [(1,9),(2,10),(3,6),(4,7),(5,8),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(51,56),(52,57),(53,58),(54,59),(55,60)], [(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)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | 30B | 30C | 30D | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 3 | 3 | 5 | 5 | 15 | 15 | 4 | 4 | 1 | 1 | 3 | 3 | 5 | 5 | 15 | 15 | 2 | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D5 | D10 | C3×D5 | C4×D5 | C6×D5 | D5×C12 | A4 | C2×A4 | C2×A4 | C2×A4 | C4×A4 | D5×A4 | C2×D5×A4 | C4×D5×A4 |
| kernel | C4×D5×A4 | A4×Dic5 | A4×C20 | C2×D5×A4 | D5×C22×C4 | D5×A4 | C22×Dic5 | C22×C20 | C23×D5 | C22×D5 | C4×A4 | C2×A4 | C22×C4 | A4 | C23 | C22 | C4×D5 | Dic5 | C20 | D10 | D5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of C4×D5×A4 ►in GL5(𝔽61)
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 0 | 50 |
| 18 | 1 | 0 | 0 | 0 |
| 42 | 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 60 | 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 36 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 45 | 1 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 13 | 6 | 0 |
| 0 | 0 | 0 | 48 | 1 |
| 0 | 0 | 0 | 14 | 0 |
G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,50,0,0,0,0,0,50,0,0,0,0,0,50],[18,42,0,0,0,1,60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,60,1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,36,0,0,0,60,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,45,0,0,0,0,1,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,13,0,0,0,0,6,48,14,0,0,0,1,0] >;
C4×D5×A4 in GAP, Magma, Sage, TeX
C_4\times D_5\times A_4
% in TeX
G:=Group("C4xD5xA4"); // GroupNames label
G:=SmallGroup(480,1036);
// by ID
G=gap.SmallGroup(480,1036);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,2,-5,92,648,271,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=b^5=c^2=d^2=e^2=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations