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## G = C5×D12order 120 = 23·3·5

### Direct product of C5 and D12

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

 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 >

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 2A 2B 2C 3 4 5A 5B 5C 5D 6 10A 10B 10C 10D 10E ··· 10L 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 30A 30B 30C 30D 60A ··· 60H order 1 2 2 2 3 4 5 5 5 5 6 10 10 10 10 10 ··· 10 12 12 15 15 15 15 20 20 20 20 30 30 30 30 60 ··· 60 size 1 1 6 6 2 2 1 1 1 1 2 1 1 1 1 6 ··· 6 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C5 C10 C10 S3 D4 D6 D12 C5×S3 C5×D4 S3×C10 C5×D12 kernel C5×D12 C60 S3×C10 D12 C12 D6 C20 C15 C10 C5 C4 C3 C2 C1 # reps 1 1 2 4 4 8 1 1 1 2 4 4 4 8

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

 4 0 0 4
,
 6 2 5 0
,
 0 2 6 0
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

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