direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C20, C22⋊C60, C23.C30, (C22×C20)⋊C3, (C22×C4)⋊C15, (C2×C10)⋊5C12, C10.4(C2×A4), C2.1(C10×A4), (C10×A4).4C2, (C2×A4).2C10, (C22×C10).2C6, SmallGroup(240,152)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C20 |
Generators and relations for A4×C20
G = < a,b,c,d | a20=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of A4×C20
►On 60 points
Generators in S60
(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 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)
(1 52 36)(2 53 37)(3 54 38)(4 55 39)(5 56 40)(6 57 21)(7 58 22)(8 59 23)(9 60 24)(10 41 25)(11 42 26)(12 43 27)(13 44 28)(14 45 29)(15 46 30)(16 47 31)(17 48 32)(18 49 33)(19 50 34)(20 51 35)
G:=sub<Sym(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), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,52,36)(2,53,37)(3,54,38)(4,55,39)(5,56,40)(6,57,21)(7,58,22)(8,59,23)(9,60,24)(10,41,25)(11,42,26)(12,43,27)(13,44,28)(14,45,29)(15,46,30)(16,47,31)(17,48,32)(18,49,33)(19,50,34)(20,51,35)>;
G:=Group( (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), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,52,36)(2,53,37)(3,54,38)(4,55,39)(5,56,40)(6,57,21)(7,58,22)(8,59,23)(9,60,24)(10,41,25)(11,42,26)(12,43,27)(13,44,28)(14,45,29)(15,46,30)(16,47,31)(17,48,32)(18,49,33)(19,50,34)(20,51,35) );
G=PermutationGroup([[(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)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60)], [(1,52,36),(2,53,37),(3,54,38),(4,55,39),(5,56,40),(6,57,21),(7,58,22),(8,59,23),(9,60,24),(10,41,25),(11,42,26),(12,43,27),(13,44,28),(14,45,29),(15,46,30),(16,47,31),(17,48,32),(18,49,33),(19,50,34),(20,51,35)]])
A4×C20 is a maximal subgroup of
C20.S4 C20.1S4 C20⋊S4
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | 12B | 12C | 12D | 15A | ··· | 15H | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30H | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C3 | C4 | C5 | C6 | C10 | C12 | C15 | C20 | C30 | C60 | A4 | C2×A4 | C4×A4 | C5×A4 | C10×A4 | A4×C20 |
| kernel | A4×C20 | C10×A4 | C22×C20 | C5×A4 | C4×A4 | C22×C10 | C2×A4 | C2×C10 | C22×C4 | A4 | C23 | C22 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of A4×C20 ►in GL3(𝔽61) generated by
| 37 | 0 | 0 |
| 0 | 37 | 0 |
| 0 | 0 | 37 |
| 60 | 0 | 0 |
| 0 | 60 | 0 |
| 16 | 13 | 1 |
| 60 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 48 | 60 |
| 0 | 1 | 0 |
| 45 | 48 | 59 |
| 12 | 46 | 13 |
G:=sub<GL(3,GF(61))| [37,0,0,0,37,0,0,0,37],[60,0,16,0,60,13,0,0,1],[60,0,0,0,1,48,0,0,60],[0,45,12,1,48,46,0,59,13] >;
A4×C20 in GAP, Magma, Sage, TeX
A_4\times C_{20} % in TeX
G:=Group("A4xC20"); // GroupNames label
G:=SmallGroup(240,152);
// by ID
G=gap.SmallGroup(240,152);
# by ID
G:=PCGroup([6,-2,-3,-5,-2,-2,2,180,1810,3251]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
Export