A4xD20 - GroupNames

G = A₄×D₂₀ order 480 = 2⁵·3·5

Direct product of A₄ and D₂₀

direct product, metabelian, soluble, monomial

Aliases: A₄×D₂₀, C₄⋊(D₅×A₄), C₅⋊₁(D₄×A₄), C₂₀⋊₁(C₂×A₄), (C₄×A₄)⋊₃D₅, (C₅×A₄)⋊₅D₄, C₂²⋊(C₃×D₂₀), (C₂²×D₂₀)⋊C₃, (A₄×C₂₀)⋊₃C₂, D₁₀⋊₁(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

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

G = A4×D20 order 480 = 25·3·5

Direct product of A4 and D20

G = A₄×D₂₀ order 480 = 2⁵·3·5

Direct product of A₄ and D₂₀