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## G = C3×C6×F5order 360 = 23·32·5

### Direct product of C3×C6 and F5

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C6×F5
G = < a,b,c,d | a3=b6=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 240 in 96 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C2×C4, C32, D5, C10, C12, C2×C6, C15, C3×C6, C3×C6, F5, D10, C2×C12, C3×D5, C30, C3×C12, C62, C2×F5, C3×C15, C3×F5, C6×D5, C6×C12, C32×D5, C3×C30, C6×F5, C32×F5, D5×C3×C6, C3×C6×F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, C3×C6, F5, C2×C12, C3×C12, C62, C2×F5, C3×F5, C6×C12, C6×F5, C32×F5, C3×C6×F5

Smallest permutation representation of C3×C6×F5
On 90 points
Generators in S90
(1 68 89)(2 69 90)(3 70 85)(4 71 86)(5 72 87)(6 67 88)(7 61 82)(8 62 83)(9 63 84)(10 64 79)(11 65 80)(12 66 81)(13 34 50)(14 35 51)(15 36 52)(16 31 53)(17 32 54)(18 33 49)(19 40 56)(20 41 57)(21 42 58)(22 37 59)(23 38 60)(24 39 55)(25 76 47)(26 77 48)(27 78 43)(28 73 44)(29 74 45)(30 75 46)
(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)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)
(1 25 37 79 36)(2 26 38 80 31)(3 27 39 81 32)(4 28 40 82 33)(5 29 41 83 34)(6 30 42 84 35)(7 49 71 73 56)(8 50 72 74 57)(9 51 67 75 58)(10 52 68 76 59)(11 53 69 77 60)(12 54 70 78 55)(13 87 45 20 62)(14 88 46 21 63)(15 89 47 22 64)(16 90 48 23 65)(17 85 43 24 66)(18 86 44 19 61)
(7 73 56 49)(8 74 57 50)(9 75 58 51)(10 76 59 52)(11 77 60 53)(12 78 55 54)(13 62 45 20)(14 63 46 21)(15 64 47 22)(16 65 48 23)(17 66 43 24)(18 61 44 19)(25 37 36 79)(26 38 31 80)(27 39 32 81)(28 40 33 82)(29 41 34 83)(30 42 35 84)

G:=sub<Sym(90)| (1,68,89)(2,69,90)(3,70,85)(4,71,86)(5,72,87)(6,67,88)(7,61,82)(8,62,83)(9,63,84)(10,64,79)(11,65,80)(12,66,81)(13,34,50)(14,35,51)(15,36,52)(16,31,53)(17,32,54)(18,33,49)(19,40,56)(20,41,57)(21,42,58)(22,37,59)(23,38,60)(24,39,55)(25,76,47)(26,77,48)(27,78,43)(28,73,44)(29,74,45)(30,75,46), (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)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90), (1,25,37,79,36)(2,26,38,80,31)(3,27,39,81,32)(4,28,40,82,33)(5,29,41,83,34)(6,30,42,84,35)(7,49,71,73,56)(8,50,72,74,57)(9,51,67,75,58)(10,52,68,76,59)(11,53,69,77,60)(12,54,70,78,55)(13,87,45,20,62)(14,88,46,21,63)(15,89,47,22,64)(16,90,48,23,65)(17,85,43,24,66)(18,86,44,19,61), (7,73,56,49)(8,74,57,50)(9,75,58,51)(10,76,59,52)(11,77,60,53)(12,78,55,54)(13,62,45,20)(14,63,46,21)(15,64,47,22)(16,65,48,23)(17,66,43,24)(18,61,44,19)(25,37,36,79)(26,38,31,80)(27,39,32,81)(28,40,33,82)(29,41,34,83)(30,42,35,84)>;

G:=Group( (1,68,89)(2,69,90)(3,70,85)(4,71,86)(5,72,87)(6,67,88)(7,61,82)(8,62,83)(9,63,84)(10,64,79)(11,65,80)(12,66,81)(13,34,50)(14,35,51)(15,36,52)(16,31,53)(17,32,54)(18,33,49)(19,40,56)(20,41,57)(21,42,58)(22,37,59)(23,38,60)(24,39,55)(25,76,47)(26,77,48)(27,78,43)(28,73,44)(29,74,45)(30,75,46), (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)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90), (1,25,37,79,36)(2,26,38,80,31)(3,27,39,81,32)(4,28,40,82,33)(5,29,41,83,34)(6,30,42,84,35)(7,49,71,73,56)(8,50,72,74,57)(9,51,67,75,58)(10,52,68,76,59)(11,53,69,77,60)(12,54,70,78,55)(13,87,45,20,62)(14,88,46,21,63)(15,89,47,22,64)(16,90,48,23,65)(17,85,43,24,66)(18,86,44,19,61), (7,73,56,49)(8,74,57,50)(9,75,58,51)(10,76,59,52)(11,77,60,53)(12,78,55,54)(13,62,45,20)(14,63,46,21)(15,64,47,22)(16,65,48,23)(17,66,43,24)(18,61,44,19)(25,37,36,79)(26,38,31,80)(27,39,32,81)(28,40,33,82)(29,41,34,83)(30,42,35,84) );

G=PermutationGroup([[(1,68,89),(2,69,90),(3,70,85),(4,71,86),(5,72,87),(6,67,88),(7,61,82),(8,62,83),(9,63,84),(10,64,79),(11,65,80),(12,66,81),(13,34,50),(14,35,51),(15,36,52),(16,31,53),(17,32,54),(18,33,49),(19,40,56),(20,41,57),(21,42,58),(22,37,59),(23,38,60),(24,39,55),(25,76,47),(26,77,48),(27,78,43),(28,73,44),(29,74,45),(30,75,46)], [(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),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90)], [(1,25,37,79,36),(2,26,38,80,31),(3,27,39,81,32),(4,28,40,82,33),(5,29,41,83,34),(6,30,42,84,35),(7,49,71,73,56),(8,50,72,74,57),(9,51,67,75,58),(10,52,68,76,59),(11,53,69,77,60),(12,54,70,78,55),(13,87,45,20,62),(14,88,46,21,63),(15,89,47,22,64),(16,90,48,23,65),(17,85,43,24,66),(18,86,44,19,61)], [(7,73,56,49),(8,74,57,50),(9,75,58,51),(10,76,59,52),(11,77,60,53),(12,78,55,54),(13,62,45,20),(14,63,46,21),(15,64,47,22),(16,65,48,23),(17,66,43,24),(18,61,44,19),(25,37,36,79),(26,38,31,80),(27,39,32,81),(28,40,33,82),(29,41,34,83),(30,42,35,84)]])

90 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4A 4B 4C 4D 5 6A ··· 6H 6I ··· 6X 10 12A ··· 12AF 15A ··· 15H 30A ··· 30H order 1 2 2 2 3 ··· 3 4 4 4 4 5 6 ··· 6 6 ··· 6 10 12 ··· 12 15 ··· 15 30 ··· 30 size 1 1 5 5 1 ··· 1 5 5 5 5 4 1 ··· 1 5 ··· 5 4 5 ··· 5 4 ··· 4 4 ··· 4

90 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 C3×C6×F5 C32×F5 D5×C3×C6 C6×F5 C32×D5 C3×C30 C3×F5 C6×D5 C3×D5 C30 C3×C6 C32 C6 C3 # reps 1 2 1 8 2 2 16 8 16 16 1 1 8 8

Matrix representation of C3×C6×F5 in GL5(𝔽61)

 13 0 0 0 0 0 47 0 0 0 0 0 47 0 0 0 0 0 47 0 0 0 0 0 47
,
 60 0 0 0 0 0 47 0 0 0 0 0 47 0 0 0 0 0 47 0 0 0 0 0 47
,
 1 0 0 0 0 0 0 0 0 60 0 1 0 0 60 0 0 1 0 60 0 0 0 1 60
,
 50 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0

G:=sub<GL(5,GF(61))| [13,0,0,0,0,0,47,0,0,0,0,0,47,0,0,0,0,0,47,0,0,0,0,0,47],[60,0,0,0,0,0,47,0,0,0,0,0,47,0,0,0,0,0,47,0,0,0,0,0,47],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,60,60,60,60],[50,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0] >;

C3×C6×F5 in GAP, Magma, Sage, TeX

C_3\times C_6\times F_5
% in TeX

G:=Group("C3xC6xF5");
// GroupNames label

G:=SmallGroup(360,145);
// by ID

G=gap.SmallGroup(360,145);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-5,216,5189,317]);
// Polycyclic

G:=Group<a,b,c,d|a^3=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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