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G = D5xD12order 240 = 24·3·5

Direct product of D5 and D12

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

Aliases: D5xD12, C20:1D6, D6:1D10, C12:4D10, D60:10C2, C60:3C22, Dic5:3D6, D10.18D6, D30:1C22, C30.12C23, C3:1(D4xD5), C4:2(S3xD5), C5:1(C2xD12), C15:1(C2xD4), (C4xD5):3S3, (C3xD5):1D4, (D5xC12):3C2, (C5xD12):3C2, C5:D12:3C2, (S3xC10):1C22, C6.12(C22xD5), C10.12(C22xS3), (C3xDic5):4C22, (C6xD5).14C22, (C2xS3xD5):1C2, C2.15(C2xS3xD5), SmallGroup(240,136)

Series: Derived Chief Lower central Upper central

C1C30 — D5xD12
C1C5C15C30C6xD5C2xS3xD5 — D5xD12
C15C30 — D5xD12
C1C2C4

Generators and relations for D5xD12
 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, C3, C4, C4, C22, C5, S3, C6, C6, C2xC4, D4, C23, D5, D5, C10, C10, C12, C12, D6, D6, C2xC6, C15, C2xD4, Dic5, C20, D10, D10, C2xC10, D12, D12, C2xC12, C22xS3, C5xS3, C3xD5, D15, C30, C4xD5, D20, C5:D4, C5xD4, C22xD5, C2xD12, C3xDic5, C60, S3xD5, C6xD5, S3xC10, D30, D4xD5, C5:D12, D5xC12, C5xD12, D60, C2xS3xD5, D5xD12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2xD4, D10, D12, C22xS3, C22xD5, C2xD12, S3xD5, D4xD5, C2xS3xD5, D5xD12

Smallest permutation representation of D5xD12
On 60 points
Generators in S60
(1 43 58 33 17)(2 44 59 34 18)(3 45 60 35 19)(4 46 49 36 20)(5 47 50 25 21)(6 48 51 26 22)(7 37 52 27 23)(8 38 53 28 24)(9 39 54 29 13)(10 40 55 30 14)(11 41 56 31 15)(12 42 57 32 16)
(1 23)(2 24)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 37)(34 38)(35 39)(36 40)(49 55)(50 56)(51 57)(52 58)(53 59)(54 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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(49 54)(50 53)(51 52)(55 60)(56 59)(57 58)

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

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

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

D5xD12 is a maximal subgroup of
D60:C4  D12:F5  C24:D10  D24:D5  Dic10:3D6  D20:D6  D60:3C4  D20:26D6  D20:29D6  S3xD4xD5  D12:14D10  D20:17D6
D5xD12 is a maximal quotient of
C24:D10  D24:D5  Dic60:C2  C24.2D10  C40.31D6  D24:7D5  D120:C2  Dic5.8D12  Dic5:4D12  D10.16D12  D10.17D12  Dic5:D12  D6:2Dic10  D60:17C4  D30:2Q8  D30:D4  D10:D12  C60:D4  C12:7D20  C20:D12  C60:Q8  D30:4D4  D30:5D4

36 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B6A6B6C10A10B10C10D10E10F12A12B12C12D15A15B20A20B30A30B60A60B60C60D
order12222222344556661010101010101212121215152020303060606060
size11556630302210222101022121212122210104444444444

36 irreducible representations

dim1111112222222224444
type+++++++++++++++++++
imageC1C2C2C2C2C2S3D4D5D6D6D6D10D10D12S3xD5D4xD5C2xS3xD5D5xD12
kernelD5xD12C5:D12D5xC12C5xD12D60C2xS3xD5C4xD5C3xD5D12Dic5C20D10C12D6D5C4C3C2C1
# reps1211121221112442224

Matrix representation of D5xD12 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] >;

D5xD12 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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