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G = D5×D12order 240 = 24·3·5

Direct product of D5 and D12

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

Aliases: D5×D12, C201D6, D61D10, C124D10, D6010C2, C603C22, Dic53D6, D10.18D6, D301C22, C30.12C23, C31(D4×D5), C42(S3×D5), C51(C2×D12), C151(C2×D4), (C4×D5)⋊3S3, (C3×D5)⋊1D4, (D5×C12)⋊3C2, (C5×D12)⋊3C2, C5⋊D123C2, (S3×C10)⋊1C22, C6.12(C22×D5), C10.12(C22×S3), (C3×Dic5)⋊4C22, (C6×D5).14C22, (C2×S3×D5)⋊1C2, C2.15(C2×S3×D5), SmallGroup(240,136)

Series: Derived Chief Lower central Upper central

C1C30 — D5×D12
C1C5C15C30C6×D5C2×S3×D5 — D5×D12
C15C30 — D5×D12
C1C2C4

Generators and relations for D5×D12
 G = < a,b,c,d | a5=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 608 in 108 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22 [×9], C5, S3 [×4], C6, C6 [×2], C2×C4, D4 [×4], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C12, C12, D6 [×2], D6 [×6], C2×C6, C15, C2×D4, Dic5, C20, D10, D10 [×6], C2×C10 [×2], D12, D12 [×3], C2×C12, C22×S3 [×2], C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, C4×D5, D20, C5⋊D4 [×2], C5×D4, C22×D5 [×2], C2×D12, C3×Dic5, C60, S3×D5 [×4], C6×D5, S3×C10 [×2], D30 [×2], D4×D5, C5⋊D12 [×2], D5×C12, C5×D12, D60, C2×S3×D5 [×2], D5×D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, C22×D5, C2×D12, S3×D5, D4×D5, C2×S3×D5, D5×D12

Smallest permutation representation of D5×D12
On 60 points
Generators in S60
(1 34 21 42 54)(2 35 22 43 55)(3 36 23 44 56)(4 25 24 45 57)(5 26 13 46 58)(6 27 14 47 59)(7 28 15 48 60)(8 29 16 37 49)(9 30 17 38 50)(10 31 18 39 51)(11 32 19 40 52)(12 33 20 41 53)
(1 60)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 37)(36 38)
(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 16)(14 15)(17 24)(18 23)(19 22)(20 21)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 46)(38 45)(39 44)(40 43)(41 42)(47 48)(49 58)(50 57)(51 56)(52 55)(53 54)(59 60)

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

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

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

D5×D12 is a maximal subgroup of
D60⋊C4  D12⋊F5  C24⋊D10  D24⋊D5  Dic103D6  D20⋊D6  D603C4  D2026D6  D2029D6  S3×D4×D5  D1214D10  D2017D6
D5×D12 is a maximal quotient of
C24⋊D10  D24⋊D5  Dic60⋊C2  C24.2D10  C40.31D6  D247D5  D120⋊C2  Dic5.8D12  Dic54D12  D10.16D12  D10.17D12  Dic5⋊D12  D62Dic10  D6017C4  D302Q8  D30⋊D4  D10⋊D12  C60⋊D4  C127D20  C20⋊D12  C60⋊Q8  D304D4  D305D4

36 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B6A6B6C10A10B10C10D10E10F12A12B12C12D15A15B20A20B30A30B60A60B60C60D
order12222222344556661010101010101212121215152020303060606060
size11556630302210222101022121212122210104444444444

36 irreducible representations

dim1111112222222224444
type+++++++++++++++++++
imageC1C2C2C2C2C2S3D4D5D6D6D6D10D10D12S3×D5D4×D5C2×S3×D5D5×D12
kernelD5×D12C5⋊D12D5×C12C5×D12D60C2×S3×D5C4×D5C3×D5D12Dic5C20D10C12D6D5C4C3C2C1
# reps1211121221112442224

Matrix representation of D5×D12 in GL4(𝔽61) generated by

1000
0100
00431
00600
,
60000
06000
00143
00060
,
233800
234600
00600
00060
,
233800
153800
00600
00060
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,43,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,43,60],[23,23,0,0,38,46,0,0,0,0,60,0,0,0,0,60],[23,15,0,0,38,38,0,0,0,0,60,0,0,0,0,60] >;

D5×D12 in GAP, Magma, Sage, TeX

D_5\times D_{12}
% in TeX

G:=Group("D5xD12");
// GroupNames label

G:=SmallGroup(240,136);
// by ID

G=gap.SmallGroup(240,136);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,116,50,490,6917]);
// Polycyclic

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

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