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G = A4xC20order 240 = 24·3·5

Direct product of C20 and A4

direct product, metabelian, soluble, monomial, A-group

Aliases: A4xC20, C22:C60, C23.C30, (C22xC20):C3, (C22xC4):C15, (C2xC10):5C12, C10.4(C2xA4), C2.1(C10xA4), (C10xA4).4C2, (C2xA4).2C10, (C22xC10).2C6, SmallGroup(240,152)

Series: Derived Chief Lower central Upper central

C1C22 — A4xC20
C1C22C23C22xC10C10xA4 — A4xC20
C22 — A4xC20
C1C20

Generators and relations for A4xC20
 G = < a,b,c,d | a20=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Subgroups: 84 in 38 conjugacy classes, 18 normal (all characteristic)
Quotients: C1, C2, C3, C4, C5, C6, C10, C12, A4, C15, C20, C2xA4, C30, C4xA4, C60, C5xA4, C10xA4, A4xC20
3C2
3C2
4C3
3C4
3C22
3C22
4C6
3C10
3C10
4C15
3C2xC4
3C2xC4
4C12
3C2xC10
3C20
3C2xC10
4C30
3C2xC20
3C2xC20
4C60

Smallest permutation representation of A4xC20
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)]])

A4xC20 is a maximal subgroup of   C20.S4  C20.1S4  C20:S4

80 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D5A5B5C5D6A6B10A10B10C10D10E···10L12A12B12C12D15A···15H20A···20H20I···20P30A···30H60A···60P
order12223344445555661010101010···101212121215···1520···2020···2030···3060···60
size113344113311114411113···344444···41···13···34···44···4

80 irreducible representations

dim111111111111333333
type++++
imageC1C2C3C4C5C6C10C12C15C20C30C60A4C2xA4C4xA4C5xA4C10xA4A4xC20
kernelA4xC20C10xA4C22xC20C5xA4C4xA4C22xC10C2xA4C2xC10C22xC4A4C23C22C20C10C5C4C2C1
# reps1122424488816112448

Matrix representation of A4xC20 in GL3(F61) generated by

3700
0370
0037
,
6000
0600
16131
,
6000
010
04860
,
010
454859
124613
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] >;

A4xC20 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

Export

Subgroup lattice of A4xC20 in TeX

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