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G = C5×D12order 120 = 23·3·5

Direct product of C5 and D12

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

Aliases: C5×D12, C156D4, C604C2, C203S3, C121C10, D61C10, C10.15D6, C30.20C22, C4⋊(C5×S3), C31(C5×D4), (S3×C10)⋊4C2, C2.4(S3×C10), C6.3(C2×C10), SmallGroup(120,23)

Series: Derived Chief Lower central Upper central

C1C6 — C5×D12
C1C3C6C30S3×C10 — C5×D12
C3C6 — C5×D12
C1C10C20

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

6C2
6C2
3C22
3C22
2S3
2S3
6C10
6C10
3D4
3C2×C10
3C2×C10
2C5×S3
2C5×S3
3C5×D4

Smallest permutation representation of C5×D12
On 60 points
Generators in S60
(1 39 60 33 14)(2 40 49 34 15)(3 41 50 35 16)(4 42 51 36 17)(5 43 52 25 18)(6 44 53 26 19)(7 45 54 27 20)(8 46 55 28 21)(9 47 56 29 22)(10 48 57 30 23)(11 37 58 31 24)(12 38 59 32 13)
(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 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)(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)| (1,39,60,33,14)(2,40,49,34,15)(3,41,50,35,16)(4,42,51,36,17)(5,43,52,25,18)(6,44,53,26,19)(7,45,54,27,20)(8,46,55,28,21)(9,47,56,29,22)(10,48,57,30,23)(11,37,58,31,24)(12,38,59,32,13), (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)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(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( (1,39,60,33,14)(2,40,49,34,15)(3,41,50,35,16)(4,42,51,36,17)(5,43,52,25,18)(6,44,53,26,19)(7,45,54,27,20)(8,46,55,28,21)(9,47,56,29,22)(10,48,57,30,23)(11,37,58,31,24)(12,38,59,32,13), (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)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(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([(1,39,60,33,14),(2,40,49,34,15),(3,41,50,35,16),(4,42,51,36,17),(5,43,52,25,18),(6,44,53,26,19),(7,45,54,27,20),(8,46,55,28,21),(9,47,56,29,22),(10,48,57,30,23),(11,37,58,31,24),(12,38,59,32,13)], [(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),(13,14),(15,24),(16,23),(17,22),(18,21),(19,20),(25,28),(26,27),(29,36),(30,35),(31,34),(32,33),(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  D125D5  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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