Copied to
clipboard

## G = F5×D12order 480 = 25·3·5

### Direct product of F5 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — F5×D12
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C6×F5 — C2×S3×F5 — F5×D12
 Lower central C15 — C30 — F5×D12
 Upper central C1 — C2 — C4

Generators and relations for F5×D12
G = < a,b,c,d | a5=b4=c12=d2=1, bab-1=a3, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1172 in 188 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, D4, C23, D5, D5, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, F5, F5, D10, D10, C2×C10, C4×S3, D12, D12, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C4×D4, C4×D5, D20, C5⋊D4, C5×D4, C2×F5, C2×F5, C22×D5, C4⋊Dic3, D6⋊C4, C4×C12, S3×C2×C4, C2×D12, C3×Dic5, C60, C3×F5, C3×F5, C3⋊F5, S3×D5, C6×D5, S3×C10, D30, C4×F5, C4⋊F5, C22⋊F5, D4×D5, C22×F5, C4×D12, C5⋊D12, D5×C12, C5×D12, D60, S3×F5, C6×F5, C2×C3⋊F5, C2×S3×D5, D4×F5, D6⋊F5, C12×F5, C60⋊C4, D5×D12, C2×S3×F5, F5×D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, F5, C4×S3, D12, C22×S3, C4×D4, C2×F5, S3×C2×C4, C2×D12, C4○D12, C22×F5, C4×D12, S3×F5, D4×F5, C2×S3×F5, F5×D12

Smallest permutation representation of F5×D12
On 60 points
Generators in S60
(1 30 59 17 47)(2 31 60 18 48)(3 32 49 19 37)(4 33 50 20 38)(5 34 51 21 39)(6 35 52 22 40)(7 36 53 23 41)(8 25 54 24 42)(9 26 55 13 43)(10 27 56 14 44)(11 28 57 15 45)(12 29 58 16 46)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 32 55 37)(14 33 56 38)(15 34 57 39)(16 35 58 40)(17 36 59 41)(18 25 60 42)(19 26 49 43)(20 27 50 44)(21 28 51 45)(22 29 52 46)(23 30 53 47)(24 31 54 48)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 23)(14 22)(15 21)(16 20)(17 19)(26 36)(27 35)(28 34)(29 33)(30 32)(37 47)(38 46)(39 45)(40 44)(41 43)(49 59)(50 58)(51 57)(52 56)(53 55)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5 6A 6B 6C 10A 10B 10C 12A 12B 12C ··· 12L 15 20 30 60A 60B order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 5 6 6 6 10 10 10 12 12 12 ··· 12 15 20 30 60 60 size 1 1 5 5 6 6 30 30 2 2 5 5 5 5 10 10 10 30 30 30 30 4 2 10 10 4 24 24 2 2 10 ··· 10 8 8 8 8 8

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 8 8 8 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D4 D6 D6 C4○D4 C4×S3 C4×S3 D12 C4○D12 F5 C2×F5 C2×F5 S3×F5 D4×F5 C2×S3×F5 F5×D12 kernel F5×D12 D6⋊F5 C12×F5 C60⋊C4 D5×D12 C2×S3×F5 C5⋊D12 C5×D12 D60 C4×F5 C3×F5 C4×D5 C2×F5 C3×D5 Dic5 C20 F5 D5 D12 C12 D6 C4 C3 C2 C1 # reps 1 2 1 1 1 2 4 2 2 1 2 1 2 2 2 2 4 4 1 1 2 1 1 1 2

Matrix representation of F5×D12 in GL8(𝔽61)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 60 0 0 0 0 1 0 0 60 0 0 0 0 0 1 0 60 0 0 0 0 0 0 1 60
,
 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 50 0 0 0 0 0 0 0 0 50 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0
,
 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 1 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60
,
 60 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 60 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,60,60,60],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60] >;

F5×D12 in GAP, Magma, Sage, TeX

F_5\times D_{12}
% in TeX

G:=Group("F5xD12");
// GroupNames label

G:=SmallGroup(480,995);
// by ID

G=gap.SmallGroup(480,995);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,219,100,1356,9414,2379]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=c^12=d^2=1,b*a*b^-1=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽