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G = C5×D12  order 120 = 23·3·5

Direct product of C5 and D12

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

Aliases: C5×D12, C15⋊6D4, C60⋊4C2, C20⋊3S3, C12⋊1C10, D6⋊1C10, C10.15D6, C30.20C22, C4⋊(C5×S3), C3⋊1(C5×D4), (S3×C10)⋊4C2, C2.4(S3×C10), C6.3(C2×C10), SmallGroup(120,23)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C5×D12
Chief series
C1 — C3 — C6 — C30 — S3×C10 — C5×D12
Lower central
C3 — C6 — C5×D12
Upper central
C1 — C10 — C20

Generators and relations for C5×D12
 G = < a,b,c | a5=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

C1
C2
6C2
6C2
C3
C5
C4
3C22
3C22
C6
2S3
2S3
C10
6C10
6C10
C15
3D4
D6
C12
D6
C20
3C2×C10
3C2×C10
C30
2C5×S3
2C5×S3
D12
3C5×D4
S3×C10
S3×C10
C60
C5×D12

Smallest permutation representation of C5×D12
►On 60 points
Generators in S60
(1 39 54 23 35)(2 40 55 24 36)(3 41 56 13 25)(4 42 57 14 26)(5 43 58 15 27)(6 44 59 16 28)(7 45 60 17 29)(8 46 49 18 30)(9 47 50 19 31)(10 48 51 20 32)(11 37 52 21 33)(12 38 53 22 34)

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

C5×D12 is a maximal subgroup of
 C15⋊D8  C5⋊D24  C20.D6  D12.D5  D12⋊D5  D12⋊5D5  C20⋊D6  C5×S3×D4

45 conjugacy classes

class 1 2A2B2C 3  4 5A5B5C5D 6 10A10B10C10D10E···10L12A12B15A15B15C15D20A20B20C20D30A30B30C30D60A···60H
order122234555561010101010···10121215151515202020203030303060···60
size1166221111211116···6222222222222222···2

45 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C5C10C10S3D4D6D12C5×S3C5×D4S3×C10C5×D12
kernelC5×D12C60S3×C10D12C12D6C20C15C10C5C4C3C2C1
# reps11244811124448

Matrix representation of C5×D12 ►in GL2(𝔽11) generated by

40
04
,
62
50
,
02
60
G:=sub<GL(2,GF(11))| [4,0,0,4],[6,5,2,0],[0,6,2,0] >;
 

C5×D12 in GAP, Magma, Sage, TeX 

C_5\times D_{12}
 
% in TeX
 
G:=Group("C5xD12");
 
// GroupNames label
 
G:=SmallGroup(120,23);
 
// by ID
 
G=gap.SmallGroup(120,23);
 
# by ID
 
G:=PCGroup([5,-2,-2,-5,-2,-3,221,106,2004]);
 
// Polycyclic
 
G:=Group<a,b,c|a^5=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C5×D12 in TeX

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