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## G = C2×C6×F5order 240 = 24·3·5

### Direct product of C2×C6 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×C6×F5
 Chief series C1 — C5 — D5 — C3×D5 — C3×F5 — C6×F5 — C2×C6×F5
 Lower central C5 — C2×C6×F5
 Upper central C1 — C2×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 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A ··· 4H 5 6A ··· 6F 6G ··· 6N 10A 10B 10C 12A ··· 12P 15A 15B 30A ··· 30F order 1 2 2 2 2 2 2 2 3 3 4 ··· 4 5 6 ··· 6 6 ··· 6 10 10 10 12 ··· 12 15 15 30 ··· 30 size 1 1 1 1 5 5 5 5 1 1 5 ··· 5 4 1 ··· 1 5 ··· 5 4 4 4 5 ··· 5 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 F5 C2×F5 C3×F5 C6×F5 kernel C2×C6×F5 C6×F5 D5×C2×C6 C22×F5 C6×D5 C2×C30 C2×F5 C22×D5 D10 C2×C10 C2×C6 C6 C22 C2 # reps 1 6 1 2 6 2 12 2 12 4 1 3 2 6

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

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

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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