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G = C2×C6×F5order 240 = 24·3·5

Direct product of C2×C6 and F5

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

Aliases: C2×C6×F5, D103C12, C10⋊(C2×C12), D5⋊(C2×C12), C5⋊(C22×C12), C303(C2×C4), (C2×C30)⋊4C4, (C6×D5)⋊7C4, (C2×C10)⋊4C12, C154(C22×C4), D5.(C22×C6), D10.7(C2×C6), (C3×D5).3C23, (C22×D5).4C6, (C6×D5).26C22, (D5×C2×C6).7C2, (C3×D5)⋊5(C2×C4), SmallGroup(240,200)

Series: Derived Chief Lower central Upper central

C1C5 — C2×C6×F5
C1C5D5C3×D5C3×F5C6×F5 — C2×C6×F5
C5 — C2×C6×F5
C1C2×C6

Generators and relations for C2×C6×F5
 G = < a,b,c,d | a2=b6=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 284 in 108 conjugacy classes, 64 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C22, C22 [×6], C5, C6 [×3], C6 [×4], C2×C4 [×6], C23, D5, D5 [×3], C10 [×3], C12 [×4], C2×C6, C2×C6 [×6], C15, C22×C4, F5 [×4], D10 [×6], C2×C10, C2×C12 [×6], C22×C6, C3×D5, C3×D5 [×3], C30 [×3], C2×F5 [×6], C22×D5, C22×C12, C3×F5 [×4], C6×D5 [×6], C2×C30, C22×F5, C6×F5 [×6], D5×C2×C6, C2×C6×F5
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, C12 [×4], C2×C6 [×7], C22×C4, F5, C2×C12 [×6], C22×C6, C2×F5 [×3], C22×C12, C3×F5, C22×F5, C6×F5 [×3], C2×C6×F5

Smallest permutation representation of C2×C6×F5
On 60 points
Generators in S60
(1 59)(2 60)(3 55)(4 56)(5 57)(6 58)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 45)(14 46)(15 47)(16 48)(17 43)(18 44)(19 36)(20 31)(21 32)(22 33)(23 34)(24 35)(25 42)(26 37)(27 38)(28 39)(29 40)(30 41)
(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 43 53 39 36)(2 44 54 40 31)(3 45 49 41 32)(4 46 50 42 33)(5 47 51 37 34)(6 48 52 38 35)(7 30 21 55 13)(8 25 22 56 14)(9 26 23 57 15)(10 27 24 58 16)(11 28 19 59 17)(12 29 20 60 18)
(1 59)(2 60)(3 55)(4 56)(5 57)(6 58)(7 32 30 45)(8 33 25 46)(9 34 26 47)(10 35 27 48)(11 36 28 43)(12 31 29 44)(13 49 21 41)(14 50 22 42)(15 51 23 37)(16 52 24 38)(17 53 19 39)(18 54 20 40)

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

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

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

C2×C6×F5 is a maximal subgroup of   D10.20D12

60 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A···4H 5 6A···6F6G···6N10A10B10C12A···12P15A15B30A···30F
order12222222334···456···66···610101012···12151530···30
size11115555115···541···15···54445···5444···4

60 irreducible representations

dim11111111114444
type+++++
imageC1C2C2C3C4C4C6C6C12C12F5C2×F5C3×F5C6×F5
kernelC2×C6×F5C6×F5D5×C2×C6C22×F5C6×D5C2×C30C2×F5C22×D5D10C2×C10C2×C6C6C22C2
# reps1612621221241326

Matrix representation of C2×C6×F5 in GL6(𝔽61)

100000
010000
0060000
0006000
0000600
0000060
,
1300000
0600000
0060000
0006000
0000600
0000060
,
100000
010000
0060606060
001000
000100
000010
,
5000000
0500000
0060000
0000060
0006000
001111

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[13,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,1,0,0,0,60,0,0,1,0,0,60,0,0,0],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,60,0,0,1,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,60,0,1] >;

C2×C6×F5 in GAP, Magma, Sage, TeX

C_2\times C_6\times F_5
% in TeX

G:=Group("C2xC6xF5");
// GroupNames label

G:=SmallGroup(240,200);
// by ID

G=gap.SmallGroup(240,200);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-5,144,3461,317]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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