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G = D5×D12order 240 = 24·3·5

Direct product of D5 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D5×D12
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — D5×D12
 Lower central C15 — C30 — D5×D12
 Upper central C1 — C2 — C4

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)])

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 6A 6B 6C 10A 10B 10C 10D 10E 10F 12A 12B 12C 12D 15A 15B 20A 20B 30A 30B 60A 60B 60C 60D order 1 2 2 2 2 2 2 2 3 4 4 5 5 6 6 6 10 10 10 10 10 10 12 12 12 12 15 15 20 20 30 30 60 60 60 60 size 1 1 5 5 6 6 30 30 2 2 10 2 2 2 10 10 2 2 12 12 12 12 2 2 10 10 4 4 4 4 4 4 4 4 4 4

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D5 D6 D6 D6 D10 D10 D12 S3×D5 D4×D5 C2×S3×D5 D5×D12 kernel D5×D12 C5⋊D12 D5×C12 C5×D12 D60 C2×S3×D5 C4×D5 C3×D5 D12 Dic5 C20 D10 C12 D6 D5 C4 C3 C2 C1 # reps 1 2 1 1 1 2 1 2 2 1 1 1 2 4 4 2 2 2 4

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

 1 0 0 0 0 1 0 0 0 0 43 1 0 0 60 0
,
 60 0 0 0 0 60 0 0 0 0 1 43 0 0 0 60
,
 23 38 0 0 23 46 0 0 0 0 60 0 0 0 0 60
,
 23 38 0 0 15 38 0 0 0 0 60 0 0 0 0 60
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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