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G = A4⋊D20order 480 = 25·3·5

The semidirect product of A4 and D20 acting via D20/D10=C2

non-abelian, soluble, monomial

Aliases: D102S4, A42D20, A4⋊C4⋊D5, (C5×A4)⋊3D4, C2.15(D5×S4), C51(A4⋊D4), C10.15(C2×S4), (C2×A4).7D10, C22⋊(C3⋊D20), (C23×D5)⋊2S3, C23.7(S3×D5), (C22×C10).7D6, (C10×A4).7C22, (C2×C5⋊S4)⋊4C2, (C2×D5×A4)⋊2C2, (C5×A4⋊C4)⋊3C2, (C2×C10)⋊2(C3⋊D4), SmallGroup(480,981)

Series: Derived Chief Lower central Upper central

C1C22C10×A4 — A4⋊D20
C1C22C2×C10C5×A4C10×A4C2×D5×A4 — A4⋊D20
C5×A4C10×A4 — A4⋊D20
C1C2

Generators and relations for A4⋊D20
 G = < a,b,c,d,e | a2=b2=c3=d20=e2=1, cac-1=dad-1=eae=ab=ba, cbc-1=a, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1196 in 124 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2 [×5], C3, C4 [×3], C22, C22 [×12], C5, S3, C6 [×2], C2×C4 [×3], D4 [×6], C23, C23 [×5], D5 [×3], C10, C10 [×2], Dic3, A4, D6, C2×C6, C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5, C20 [×2], D10, D10 [×9], C2×C10, C2×C10 [×2], C3⋊D4, S4, C2×A4, C2×A4, C3×D5, D15, C30, C22≀C2, D20 [×4], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C22×D5 [×5], C22×C10, A4⋊C4, C2×S4, C22×A4, C5×Dic3, C5×A4, C6×D5, D30, D10⋊C4 [×2], C5×C22⋊C4, C2×D20 [×2], C2×C5⋊D4, C23×D5, A4⋊D4, C3⋊D20, C5⋊S4, D5×A4, C10×A4, C22⋊D20, C5×A4⋊C4, C2×C5⋊S4, C2×D5×A4, A4⋊D20
Quotients: C1, C2 [×3], C22, S3, D4, D5, D6, D10, C3⋊D4, S4, D20, C2×S4, S3×D5, A4⋊D4, C3⋊D20, D5×S4, A4⋊D20

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

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

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

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

34 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C5A5B6A6B6C10A10B10C10D10E10F15A15B20A···20H30A30B
order1222222344455666101010101010151520···203030
size113310306081212602284040226666161612···121616

34 irreducible representations

dim111122222223344666
type+++++++++++++++++
imageC1C2C2C2S3D4D5D6D10C3⋊D4D20S4C2×S4S3×D5C3⋊D20A4⋊D4D5×S4A4⋊D20
kernelA4⋊D20C5×A4⋊C4C2×C5⋊S4C2×D5×A4C23×D5C5×A4A4⋊C4C22×C10C2×A4C2×C10A4D10C10C23C22C5C2C1
# reps111111212242222144

Matrix representation of A4⋊D20 in GL5(𝔽61)

10000
01000
000601
000600
001600
,
10000
01000
000160
001060
000060
,
10000
01000
00001
00100
00010
,
4652000
946000
000600
006000
000060
,
060000
600000
00010
00100
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],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,60,60,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[46,9,0,0,0,52,46,0,0,0,0,0,0,60,0,0,0,60,0,0,0,0,0,0,60],[0,60,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

A4⋊D20 in GAP, Magma, Sage, TeX

A_4\rtimes D_{20}
% in TeX

G:=Group("A4:D20");
// GroupNames label

G:=SmallGroup(480,981);
// by ID

G=gap.SmallGroup(480,981);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,85,36,234,3364,5052,1286,2953,2232]);
// Polycyclic

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

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