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## G = A4×C20order 240 = 24·3·5

### Direct product of C20 and A4

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

 Derived series C1 — C22 — A4×C20
 Chief series C1 — C22 — C23 — C22×C10 — C10×A4 — A4×C20
 Lower central C22 — A4×C20
 Upper central C1 — 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

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