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G = D5×C12order 120 = 23·3·5

Direct product of C12 and D5

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

Aliases: D5×C12, C202C6, C605C2, C12Dic5, Dic52C6, D10.2C6, C6.14D10, C30.14C22, C52(C2×C12), C158(C2×C4), C4(C3×Dic5), C2.1(C6×D5), C10.2(C2×C6), C12(C3×Dic5), (C6×D5).4C2, (C3×Dic5)⋊5C2, SmallGroup(120,17)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C12
C1C5C10C30C6×D5 — D5×C12
C5 — D5×C12
C1C12

Generators and relations for D5×C12
 G = < a,b,c | a12=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C2
5C4
5C22
5C6
5C6
5C2×C4
5C12
5C2×C6
5C2×C12

Smallest permutation representation of D5×C12
On 60 points
Generators in S60
(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 50 39 34 20)(2 51 40 35 21)(3 52 41 36 22)(4 53 42 25 23)(5 54 43 26 24)(6 55 44 27 13)(7 56 45 28 14)(8 57 46 29 15)(9 58 47 30 16)(10 59 48 31 17)(11 60 37 32 18)(12 49 38 33 19)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 49)(34 50)(35 51)(36 52)

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

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

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

D5×C12 is a maximal subgroup of   C20.32D6  C60.C4  C12.F5  C60⋊C4  D6.D10  D125D5  C12.28D10

48 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D5A5B6A6B6C6D6E6F10A10B12A12B12C12D12E12F12G12H15A15B15C15D20A20B20C20D30A30B30C30D60A···60H
order1222334444556666661010121212121212121215151515202020203030303060···60
size11551111552211555522111155552222222222222···2

48 irreducible representations

dim1111111111222222
type++++++
imageC1C2C2C2C3C4C6C6C6C12D5D10C3×D5C4×D5C6×D5D5×C12
kernelD5×C12C3×Dic5C60C6×D5C4×D5C3×D5Dic5C20D10D5C12C6C4C3C2C1
# reps1111242228224448

Matrix representation of D5×C12 in GL2(𝔽61) generated by

210
021
,
01
6017
,
10
1760
G:=sub<GL(2,GF(61))| [21,0,0,21],[0,60,1,17],[1,17,0,60] >;

D5×C12 in GAP, Magma, Sage, TeX

D_5\times C_{12}
% in TeX

G:=Group("D5xC12");
// GroupNames label

G:=SmallGroup(120,17);
// by ID

G=gap.SmallGroup(120,17);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-5,66,2404]);
// Polycyclic

G:=Group<a,b,c|a^12=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D5×C12 in TeX

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