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G = C3×D20order 120 = 23·3·5

Direct product of C3 and D20

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

Aliases: C3×D20, C155D4, C201C6, C603C2, C123D5, D101C6, C6.15D10, C30.15C22, C4⋊(C3×D5), C51(C3×D4), (C6×D5)⋊4C2, C2.4(C6×D5), C10.3(C2×C6), SmallGroup(120,18)

Series: Derived Chief Lower central Upper central

C1C10 — C3×D20
C1C5C10C30C6×D5 — C3×D20
C5C10 — C3×D20
C1C6C12

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

10C2
10C2
5C22
5C22
10C6
10C6
2D5
2D5
5D4
5C2×C6
5C2×C6
2C3×D5
2C3×D5
5C3×D4

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

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

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

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

C3×D20 is a maximal subgroup of
C15⋊D8  C3⋊D40  C30.D4  C6.D20  D205S3  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

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