Copied to
clipboard

G = C3×D20  order 120 = 23·3·5

Direct product of C3 and D20

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C3×D20, C15⋊5D4, C20⋊1C6, C60⋊3C2, C12⋊3D5, D10⋊1C6, C6.15D10, C30.15C22, C4⋊(C3×D5), C5⋊1(C3×D4), (C6×D5)⋊4C2, C2.4(C6×D5), C10.3(C2×C6), SmallGroup(120,18)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C3×D20
Chief series
C1 — C5 — C10 — C30 — C6×D5 — C3×D20
Lower central
C5 — C10 — C3×D20
Upper central
C1 — C6 — C12

Generators and relations for C3×D20
 G = < a,b,c | a3=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >

C1
C2
10C2
10C2
C3
C5
C4
5C22
5C22
C6
10C6
10C6
C10
2D5
2D5
C15
5D4
C12
5C2×C6
5C2×C6
D10
C20
D10
C30
2C3×D5
2C3×D5
5C3×D4
D20
C6×D5
C6×D5
C60
C3×D20

Smallest permutation representation of C3×D20
►On 60 points
Generators in S60
(1 25 47)(2 26 48)(3 27 49)(4 28 50)(5 29 51)(6 30 52)(7 31 53)(8 32 54)(9 33 55)(10 34 56)(11 35 57)(12 36 58)(13 37 59)(14 38 60)(15 39 41)(16 40 42)(17 21 43)(18 22 44)(19 23 45)(20 24 46)

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

C3×D20 is a maximal subgroup of
 C15⋊D8  C3⋊D40  C30.D4  C6.D20  D20⋊5S3  D20⋊S3  C20⋊D6  C3×D4×D5  D20.A4

39 conjugacy classes

class 1 2A2B2C3A3B 4 5A5B6A6B6C6D6E6F10A10B12A12B15A15B15C15D20A20B20C20D30A30B30C30D60A···60H
order1222334556666661010121215151515202020203030303060···60
size11101011222111010101022222222222222222···2

39 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C3C6C6D4D5D10C3×D4C3×D5D20C6×D5C3×D20
kernelC3×D20C60C6×D5D20C20D10C15C12C6C5C4C3C2C1
# reps11222412224448

Matrix representation of C3×D20 ►in GL2(𝔽19) generated by

110
011
,
67
80
,
07
110
G:=sub<GL(2,GF(19))| [11,0,0,11],[6,8,7,0],[0,11,7,0] >;
 

C3×D20 in GAP, Magma, Sage, TeX 

C_3\times D_{20}
 
% in TeX
 
G:=Group("C3xD20");
 
// GroupNames label
 
G:=SmallGroup(120,18);
 
// by ID
 
G=gap.SmallGroup(120,18);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-2,-5,141,66,2404]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C3×D20 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁