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G = C3xC6xF5order 360 = 23·32·5

Direct product of C3xC6 and F5

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C3xC6xF5, C30:2C12, D5.C62, C5:(C6xC12), C10:(C3xC12), D5:(C3xC12), (C3xC30):4C4, D10.(C3xC6), C15:3(C2xC12), (C3xD5):3C12, (C6xD5).6C6, (C32xD5):7C4, (C32xD5).7C22, (D5xC3xC6).5C2, (C3xC15):10(C2xC4), (C3xD5).4(C2xC6), SmallGroup(360,145)

Series: Derived Chief Lower central Upper central

C1C5 — C3xC6xF5
C1C5D5C3xD5C32xD5C32xF5 — C3xC6xF5
C5 — C3xC6xF5
C1C3xC6

Generators and relations for C3xC6xF5
 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, C2xC4, C32, D5, C10, C12, C2xC6, C15, C3xC6, C3xC6, F5, D10, C2xC12, C3xD5, C30, C3xC12, C62, C2xF5, C3xC15, C3xF5, C6xD5, C6xC12, C32xD5, C3xC30, C6xF5, C32xF5, D5xC3xC6, C3xC6xF5
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, C32, C12, C2xC6, C3xC6, F5, C2xC12, C3xC12, C62, C2xF5, C3xF5, C6xC12, C6xF5, C32xF5, C3xC6xF5

Smallest permutation representation of C3xC6xF5
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 2A2B2C3A···3H4A4B4C4D 5 6A···6H6I···6X 10 12A···12AF15A···15H30A···30H
order12223···3444456···66···61012···1215···1530···30
size11551···1555541···15···545···54···44···4

90 irreducible representations

dim11111111114444
type+++++
imageC1C2C2C3C4C4C6C6C12C12F5C2xF5C3xF5C6xF5
kernelC3xC6xF5C32xF5D5xC3xC6C6xF5C32xD5C3xC30C3xF5C6xD5C3xD5C30C3xC6C32C6C3
# reps12182216816161188

Matrix representation of C3xC6xF5 in GL5(F61)

130000
047000
004700
000470
000047
,
600000
047000
004700
000470
000047
,
10000
000060
010060
001060
000160
,
500000
00010
01000
00001
00100

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] >;

C3xC6xF5 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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