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## G = A4×D20order 480 = 25·3·5

### Direct product of A4 and D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — A4×D20
 Chief series C1 — C5 — C2×C10 — C22×C10 — C10×A4 — C2×D5×A4 — A4×D20
 Lower central C2×C10 — C22×C10 — A4×D20
 Upper central C1 — C2 — C4

Generators and relations for A4×D20
G = < a,b,c,d,e | a2=b2=c3=d20=e2=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1080 in 132 conjugacy classes, 27 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C2×C4, D4, C23, C23, D5, C10, C10, C12, A4, C2×C6, C15, C22×C4, C2×D4, C24, C20, C20, D10, D10, C2×C10, C2×C10, C3×D4, C2×A4, C2×A4, C3×D5, C30, C22×D4, D20, D20, C2×C20, C22×D5, C22×C10, C4×A4, C22×A4, C60, C5×A4, C6×D5, C2×D20, C22×C20, C23×D5, D4×A4, C3×D20, D5×A4, C10×A4, C22×D20, A4×C20, C2×D5×A4, A4×D20
Quotients: C1, C2, C3, C22, C6, D4, D5, A4, C2×C6, D10, C3×D4, C2×A4, C3×D5, D20, C22×A4, C6×D5, D4×A4, C3×D20, D5×A4, C2×D5×A4, A4×D20

Smallest permutation representation of A4×D20
On 60 points
Generators in S60
(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)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)
(1 44 28)(2 45 29)(3 46 30)(4 47 31)(5 48 32)(6 49 33)(7 50 34)(8 51 35)(9 52 36)(10 53 37)(11 54 38)(12 55 39)(13 56 40)(14 57 21)(15 58 22)(16 59 23)(17 60 24)(18 41 25)(19 42 26)(20 43 27)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 46)(42 45)(43 44)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)

G:=sub<Sym(60)| (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), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,44,28)(2,45,29)(3,46,30)(4,47,31)(5,48,32)(6,49,33)(7,50,34)(8,51,35)(9,52,36)(10,53,37)(11,54,38)(12,55,39)(13,56,40)(14,57,21)(15,58,22)(16,59,23)(17,60,24)(18,41,25)(19,42,26)(20,43,27), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,46)(42,45)(43,44)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)>;

G:=Group( (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), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,44,28)(2,45,29)(3,46,30)(4,47,31)(5,48,32)(6,49,33)(7,50,34)(8,51,35)(9,52,36)(10,53,37)(11,54,38)(12,55,39)(13,56,40)(14,57,21)(15,58,22)(16,59,23)(17,60,24)(18,41,25)(19,42,26)(20,43,27), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,46)(42,45)(43,44)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54) );

G=PermutationGroup([[(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)], [(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60)], [(1,44,28),(2,45,29),(3,46,30),(4,47,31),(5,48,32),(6,49,33),(7,50,34),(8,51,35),(9,52,36),(10,53,37),(11,54,38),(12,55,39),(13,56,40),(14,57,21),(15,58,22),(16,59,23),(17,60,24),(18,41,25),(19,42,26),(20,43,27)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(35,40),(36,39),(37,38),(41,46),(42,45),(43,44),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54)]])

52 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 5A 5B 6A 6B 6C 6D 6E 6F 10A 10B 10C 10D 10E 10F 12A 12B 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 5 5 6 6 6 6 6 6 10 10 10 10 10 10 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 10 10 30 30 4 4 2 6 2 2 4 4 40 40 40 40 2 2 6 6 6 6 8 8 8 8 8 8 2 2 2 2 6 6 6 6 8 8 8 8 8 ··· 8

52 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 6 6 6 6 type + + + + + + + + + + + + + + image C1 C2 C2 C3 C6 C6 D4 D5 D10 C3×D4 C3×D5 D20 C6×D5 C3×D20 A4 C2×A4 C2×A4 D4×A4 D5×A4 C2×D5×A4 A4×D20 kernel A4×D20 A4×C20 C2×D5×A4 C22×D20 C22×C20 C23×D5 C5×A4 C4×A4 C2×A4 C2×C10 C22×C4 A4 C23 C22 D20 C20 D10 C5 C4 C2 C1 # reps 1 1 2 2 2 4 1 2 2 2 4 4 4 8 1 1 2 1 2 2 4

Matrix representation of A4×D20 in GL5(𝔽61)

 1 0 0 0 0 0 1 0 0 0 0 0 0 60 1 0 0 0 60 0 0 0 1 60 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 60 0 1 0 0 60 1 0
,
 13 0 0 0 0 0 13 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 26 22 0 0 0 39 26 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 39 26 0 0 0 26 22 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,60,60,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,60,60,60,0,0,0,0,1,0,0,0,1,0],[13,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[26,39,0,0,0,22,26,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[39,26,0,0,0,26,22,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4×D20 in GAP, Magma, Sage, TeX

A_4\times D_{20}
% in TeX

G:=Group("A4xD20");
// GroupNames label

G:=SmallGroup(480,1037);
// by ID

G=gap.SmallGroup(480,1037);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-5,197,92,648,271,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^20=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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