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G = C3xD20  order 120 = 23·3·5

Direct product of C3 and D20

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

Aliases: C3xD20, C15:5D4, C20:1C6, C60:3C2, C12:3D5, D10:1C6, C6.15D10, C30.15C22, C4:(C3xD5), C5:1(C3xD4), (C6xD5):4C2, C2.4(C6xD5), C10.3(C2xC6), SmallGroup(120,18)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C3xD20
Chief series
C1 — C5 — C10 — C30 — C6xD5 — C3xD20
Lower central
C5 — C10 — C3xD20
Upper central
C1 — C6 — C12

Generators and relations for C3xD20
 G = < a,b,c | a3=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 96 in 32 conjugacy classes, 18 normal (14 characteristic)
Quotients: C1, C2, C3, C22, C6, D4, D5, C2xC6, D10, C3xD4, C3xD5, D20, C6xD5, C3xD20
C1
C2
10C2
10C2
C3
C5
C4
5C22
5C22
C6
10C6
10C6
C10
2D5
2D5
C15
5D4
C12
5C2xC6
5C2xC6
D10
C20
D10
C30
2C3xD5
2C3xD5
5C3xD4
D20
C6xD5
C6xD5
C60
C3xD20

Smallest permutation representation of C3xD20
►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)]])
 

C3xD20 is a maximal subgroup of
 C15:D8  C3:D40  C30.D4  C6.D20  D20:5S3  D20:S3  C20:D6  C3xD4xD5  D20.A4

39 conjugacy classes

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

39 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C3C6C6D4D5D10C3xD4C3xD5D20C6xD5C3xD20
kernelC3xD20C60C6xD5D20C20D10C15C12C6C5C4C3C2C1
# reps11222412224448

Matrix representation of C3xD20 ►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] >;
 

C3xD20 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
 

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Subgroup lattice of C3xD20 in TeX

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