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## G = C6×C3⋊F5order 360 = 23·32·5

### Direct product of C6 and C3⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C6×C3⋊F5
 Chief series C1 — C5 — C15 — C3×D5 — C32×D5 — C3×C3⋊F5 — C6×C3⋊F5
 Lower central C15 — C6×C3⋊F5
 Upper central C1 — C6

Generators and relations for C6×C3⋊F5
G = < a,b,c,d | a6=b3=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 276 in 74 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, C6, C6, C2×C4, C32, D5, C10, Dic3, C12, C2×C6, C15, C15, C3×C6, C3×C6, F5, D10, C2×Dic3, C2×C12, C3×D5, C3×D5, C30, C30, C3×Dic3, C62, C2×F5, C3×C15, C3×F5, C3⋊F5, C6×D5, C6×D5, C6×Dic3, C32×D5, C3×C30, C6×F5, C2×C3⋊F5, C3×C3⋊F5, D5×C3×C6, C6×C3⋊F5
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, C3×S3, F5, C2×Dic3, C2×C12, C3×Dic3, S3×C6, C2×F5, C3×F5, C3⋊F5, C6×Dic3, C6×F5, C2×C3⋊F5, C3×C3⋊F5, C6×C3⋊F5

Smallest permutation representation of C6×C3⋊F5
On 60 points
Generators in S60
(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 5)(2 4 6)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 59 57)(56 60 58)
(1 48 35 51 42)(2 43 36 52 37)(3 44 31 53 38)(4 45 32 54 39)(5 46 33 49 40)(6 47 34 50 41)(7 58 26 19 14)(8 59 27 20 15)(9 60 28 21 16)(10 55 29 22 17)(11 56 30 23 18)(12 57 25 24 13)
(1 16 4 13)(2 17 5 14)(3 18 6 15)(7 36 22 49)(8 31 23 50)(9 32 24 51)(10 33 19 52)(11 34 20 53)(12 35 21 54)(25 48 60 39)(26 43 55 40)(27 44 56 41)(28 45 57 42)(29 46 58 37)(30 47 59 38)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 5 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J ··· 6O 10 12A ··· 12H 15A ··· 15H 30A ··· 30H order 1 2 2 2 3 3 3 3 3 4 4 4 4 5 6 6 6 6 6 6 6 6 6 6 ··· 6 10 12 ··· 12 15 ··· 15 30 ··· 30 size 1 1 5 5 1 1 2 2 2 15 15 15 15 4 1 1 2 2 2 5 5 5 5 10 ··· 10 4 15 ··· 15 4 ··· 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + - + - + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 S3 Dic3 D6 Dic3 C3×S3 C3×Dic3 S3×C6 C3×Dic3 F5 C2×F5 C3×F5 C3⋊F5 C6×F5 C2×C3⋊F5 C3×C3⋊F5 C6×C3⋊F5 kernel C6×C3⋊F5 C3×C3⋊F5 D5×C3×C6 C2×C3⋊F5 C32×D5 C3×C30 C3⋊F5 C6×D5 C3×D5 C30 C6×D5 C3×D5 C3×D5 C30 D10 D5 D5 C10 C3×C6 C32 C6 C6 C3 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 4 4 1 1 1 1 2 2 2 2 1 1 2 2 2 2 4 4

Matrix representation of C6×C3⋊F5 in GL4(𝔽61) generated by

 48 0 0 0 0 48 0 0 0 0 48 0 0 0 0 48
,
 47 0 0 0 0 47 0 0 0 0 13 0 0 0 0 13
,
 0 1 0 0 60 17 0 0 0 0 44 44 0 0 17 60
,
 0 0 1 0 0 0 17 60 60 0 0 0 0 60 0 0
G:=sub<GL(4,GF(61))| [48,0,0,0,0,48,0,0,0,0,48,0,0,0,0,48],[47,0,0,0,0,47,0,0,0,0,13,0,0,0,0,13],[0,60,0,0,1,17,0,0,0,0,44,17,0,0,44,60],[0,0,60,0,0,0,0,60,1,17,0,0,0,60,0,0] >;

C6×C3⋊F5 in GAP, Magma, Sage, TeX

C_6\times C_3\rtimes F_5
% in TeX

G:=Group("C6xC3:F5");
// GroupNames label

G:=SmallGroup(360,146);
// by ID

G=gap.SmallGroup(360,146);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,1444,7781,887]);
// Polycyclic

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

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