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G = A4×D20order 480 = 25·3·5

Direct product of A4 and D20

direct product, metabelian, soluble, monomial

Aliases: A4×D20, C4⋊(D5×A4), C51(D4×A4), C201(C2×A4), (C4×A4)⋊3D5, (C5×A4)⋊5D4, C22⋊(C3×D20), (C22×D20)⋊C3, (A4×C20)⋊3C2, D101(C2×A4), (C22×C20)⋊1C6, (C23×D5)⋊1C6, (C2×A4).15D10, C10.3(C22×A4), C23.12(C6×D5), (C10×A4).15C22, (C2×D5×A4)⋊4C2, C2.4(C2×D5×A4), (C22×C4)⋊(C3×D5), (C2×C10)⋊1(C3×D4), (C22×C10).3(C2×C6), SmallGroup(480,1037)

Series: Derived Chief Lower central Upper central

C1C22×C10 — A4×D20
C1C5C2×C10C22×C10C10×A4C2×D5×A4 — A4×D20
C2×C10C22×C10 — A4×D20
C1C2C4

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 2A2B2C2D2E2F2G3A3B4A4B5A5B6A6B6C6D6E6F10A10B10C10D10E10F12A12B15A15B15C15D20A20B20C20D20E20F20G20H30A30B30C30D60A···60H
order1222222233445566666610101010101012121515151520202020202020203030303060···60
size11331010303044262244404040402266668888882222666688888···8

52 irreducible representations

dim111111222222223336666
type++++++++++++++
imageC1C2C2C3C6C6D4D5D10C3×D4C3×D5D20C6×D5C3×D20A4C2×A4C2×A4D4×A4D5×A4C2×D5×A4A4×D20
kernelA4×D20A4×C20C2×D5×A4C22×D20C22×C20C23×D5C5×A4C4×A4C2×A4C2×C10C22×C4A4C23C22D20C20D10C5C4C2C1
# reps112224122244481121224

Matrix representation of A4×D20 in GL5(𝔽61)

10000
01000
000601
000600
001600
,
10000
01000
006000
006001
006010
,
130000
013000
00010
00001
00100
,
2622000
3926000
00100
00010
00001
,
3926000
2622000
00100
00010
00001

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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