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G = F5×C22×C6  order 480 = 25·3·5

Direct product of C22×C6 and F5

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

Aliases: F5×C22×C6, C5⋊(C23×C12), C10⋊(C22×C12), C15⋊4(C23×C4), D5⋊(C22×C12), D5.(C23×C6), D10⋊9(C2×C12), C30⋊4(C22×C4), (C22×C30)⋊9C4, (C22×C10)⋊7C12, (C22×D5)⋊8C12, (C23×D5).6C6, (C3×D5).3C24, (C6×D5).73C23, D10.14(C22×C6), (D5×C2×C6)⋊13C4, (C2×C30)⋊10(C2×C4), (C2×C10)⋊6(C2×C12), (C6×D5)⋊34(C2×C4), (D5×C22×C6).9C2, (C3×D5)⋊5(C22×C4), (D5×C2×C6).153C22, (C22×D5).42(C2×C6), SmallGroup(480,1205)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — F5×C22×C6
Chief series
C1 — C5 — D5 — C3×D5 — C3×F5 — C6×F5 — C2×C6×F5 — F5×C22×C6
Lower central
C5 — F5×C22×C6

Subgroups: 1288 in 472 conjugacy classes, 268 normal (14 characteristic)
C1, C2 [×7], C2 [×8], C3, C4 [×8], C22 [×7], C22 [×28], C5, C6 [×7], C6 [×8], C2×C4 [×28], C23, C23 [×14], D5, D5 [×7], C10 [×7], C12 [×8], C2×C6 [×7], C2×C6 [×28], C15, C22×C4 [×14], C24, F5 [×8], D10 [×28], C2×C10 [×7], C2×C12 [×28], C22×C6, C22×C6 [×14], C3×D5, C3×D5 [×7], C30 [×7], C23×C4, C2×F5 [×28], C22×D5 [×14], C22×C10, C22×C12 [×14], C23×C6, C3×F5 [×8], C6×D5 [×28], C2×C30 [×7], C22×F5 [×14], C23×D5, C23×C12, C6×F5 [×28], D5×C2×C6 [×14], C22×C30, C23×F5, C2×C6×F5 [×14], D5×C22×C6, F5×C22×C6

Quotients:
C1, C2 [×15], C3, C4 [×8], C22 [×35], C6 [×15], C2×C4 [×28], C23 [×15], C12 [×8], C2×C6 [×35], C22×C4 [×14], C24, F5, C2×C12 [×28], C22×C6 [×15], C23×C4, C2×F5 [×7], C22×C12 [×14], C23×C6, C3×F5, C22×F5 [×7], C23×C12, C6×F5 [×7], C23×F5, C2×C6×F5 [×7], F5×C22×C6

Generators and relations
 G = < a,b,c,d,e | a2=b2=c6=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Smallest permutation representation
►On 120 points
Generators in S120
(1 86)(2 87)(3 88)(4 89)(5 90)(6 85)(7 77)(8 78)(9 73)(10 74)(11 75)(12 76)(13 83)(14 84)(15 79)(16 80)(17 81)(18 82)(19 63)(20 64)(21 65)(22 66)(23 61)(24 62)(25 69)(26 70)(27 71)(28 72)(29 67)(30 68)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)

(1 59)(2 60)(3 55)(4 56)(5 57)(6 58)(7 104)(8 105)(9 106)(10 107)(11 108)(12 103)(13 110)(14 111)(15 112)(16 113)(17 114)(18 109)(19 96)(20 91)(21 92)(22 93)(23 94)(24 95)(25 102)(26 97)(27 98)(28 99)(29 100)(30 101)(31 64)(32 65)(33 66)(34 61)(35 62)(36 63)(37 70)(38 71)(39 72)(40 67)(41 68)(42 69)(43 76)(44 77)(45 78)(46 73)(47 74)(48 75)(49 82)(50 83)(51 84)(52 79)(53 80)(54 81)(85 118)(86 119)(87 120)(88 115)(89 116)(90 117)

(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)(91 92 93 94 95 96)(97 98 99 100 101 102)(103 104 105 106 107 108)(109 110 111 112 113 114)(115 116 117 118 119 120)

(1 38 53 44 36)(2 39 54 45 31)(3 40 49 46 32)(4 41 50 47 33)(5 42 51 48 34)(6 37 52 43 35)(7 19 119 27 16)(8 20 120 28 17)(9 21 115 29 18)(10 22 116 30 13)(11 23 117 25 14)(12 24 118 26 15)(55 67 82 73 65)(56 68 83 74 66)(57 69 84 75 61)(58 70 79 76 62)(59 71 80 77 63)(60 72 81 78 64)(85 97 112 103 95)(86 98 113 104 96)(87 99 114 105 91)(88 100 109 106 92)(89 101 110 107 93)(90 102 111 108 94)

(1 89)(2 90)(3 85)(4 86)(5 87)(6 88)(7 68 16 66)(8 69 17 61)(9 70 18 62)(10 71 13 63)(11 72 14 64)(12 67 15 65)(19 74 27 83)(20 75 28 84)(21 76 29 79)(22 77 30 80)(23 78 25 81)(24 73 26 82)(31 108 39 111)(32 103 40 112)(33 104 41 113)(34 105 42 114)(35 106 37 109)(36 107 38 110)(43 100 52 92)(44 101 53 93)(45 102 54 94)(46 97 49 95)(47 98 50 96)(48 99 51 91)(55 118)(56 119)(57 120)(58 115)(59 116)(60 117)
 
G:=sub<Sym(120)| (1,86)(2,87)(3,88)(4,89)(5,90)(6,85)(7,77)(8,78)(9,73)(10,74)(11,75)(12,76)(13,83)(14,84)(15,79)(16,80)(17,81)(18,82)(19,63)(20,64)(21,65)(22,66)(23,61)(24,62)(25,69)(26,70)(27,71)(28,72)(29,67)(30,68)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120), (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,104)(8,105)(9,106)(10,107)(11,108)(12,103)(13,110)(14,111)(15,112)(16,113)(17,114)(18,109)(19,96)(20,91)(21,92)(22,93)(23,94)(24,95)(25,102)(26,97)(27,98)(28,99)(29,100)(30,101)(31,64)(32,65)(33,66)(34,61)(35,62)(36,63)(37,70)(38,71)(39,72)(40,67)(41,68)(42,69)(43,76)(44,77)(45,78)(46,73)(47,74)(48,75)(49,82)(50,83)(51,84)(52,79)(53,80)(54,81)(85,118)(86,119)(87,120)(88,115)(89,116)(90,117), (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)(91,92,93,94,95,96)(97,98,99,100,101,102)(103,104,105,106,107,108)(109,110,111,112,113,114)(115,116,117,118,119,120), (1,38,53,44,36)(2,39,54,45,31)(3,40,49,46,32)(4,41,50,47,33)(5,42,51,48,34)(6,37,52,43,35)(7,19,119,27,16)(8,20,120,28,17)(9,21,115,29,18)(10,22,116,30,13)(11,23,117,25,14)(12,24,118,26,15)(55,67,82,73,65)(56,68,83,74,66)(57,69,84,75,61)(58,70,79,76,62)(59,71,80,77,63)(60,72,81,78,64)(85,97,112,103,95)(86,98,113,104,96)(87,99,114,105,91)(88,100,109,106,92)(89,101,110,107,93)(90,102,111,108,94), (1,89)(2,90)(3,85)(4,86)(5,87)(6,88)(7,68,16,66)(8,69,17,61)(9,70,18,62)(10,71,13,63)(11,72,14,64)(12,67,15,65)(19,74,27,83)(20,75,28,84)(21,76,29,79)(22,77,30,80)(23,78,25,81)(24,73,26,82)(31,108,39,111)(32,103,40,112)(33,104,41,113)(34,105,42,114)(35,106,37,109)(36,107,38,110)(43,100,52,92)(44,101,53,93)(45,102,54,94)(46,97,49,95)(47,98,50,96)(48,99,51,91)(55,118)(56,119)(57,120)(58,115)(59,116)(60,117)>;
 
G:=Group( (1,86)(2,87)(3,88)(4,89)(5,90)(6,85)(7,77)(8,78)(9,73)(10,74)(11,75)(12,76)(13,83)(14,84)(15,79)(16,80)(17,81)(18,82)(19,63)(20,64)(21,65)(22,66)(23,61)(24,62)(25,69)(26,70)(27,71)(28,72)(29,67)(30,68)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120), (1,59)(2,60)(3,55)(4,56)(5,57)(6,58)(7,104)(8,105)(9,106)(10,107)(11,108)(12,103)(13,110)(14,111)(15,112)(16,113)(17,114)(18,109)(19,96)(20,91)(21,92)(22,93)(23,94)(24,95)(25,102)(26,97)(27,98)(28,99)(29,100)(30,101)(31,64)(32,65)(33,66)(34,61)(35,62)(36,63)(37,70)(38,71)(39,72)(40,67)(41,68)(42,69)(43,76)(44,77)(45,78)(46,73)(47,74)(48,75)(49,82)(50,83)(51,84)(52,79)(53,80)(54,81)(85,118)(86,119)(87,120)(88,115)(89,116)(90,117), (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)(91,92,93,94,95,96)(97,98,99,100,101,102)(103,104,105,106,107,108)(109,110,111,112,113,114)(115,116,117,118,119,120), (1,38,53,44,36)(2,39,54,45,31)(3,40,49,46,32)(4,41,50,47,33)(5,42,51,48,34)(6,37,52,43,35)(7,19,119,27,16)(8,20,120,28,17)(9,21,115,29,18)(10,22,116,30,13)(11,23,117,25,14)(12,24,118,26,15)(55,67,82,73,65)(56,68,83,74,66)(57,69,84,75,61)(58,70,79,76,62)(59,71,80,77,63)(60,72,81,78,64)(85,97,112,103,95)(86,98,113,104,96)(87,99,114,105,91)(88,100,109,106,92)(89,101,110,107,93)(90,102,111,108,94), (1,89)(2,90)(3,85)(4,86)(5,87)(6,88)(7,68,16,66)(8,69,17,61)(9,70,18,62)(10,71,13,63)(11,72,14,64)(12,67,15,65)(19,74,27,83)(20,75,28,84)(21,76,29,79)(22,77,30,80)(23,78,25,81)(24,73,26,82)(31,108,39,111)(32,103,40,112)(33,104,41,113)(34,105,42,114)(35,106,37,109)(36,107,38,110)(43,100,52,92)(44,101,53,93)(45,102,54,94)(46,97,49,95)(47,98,50,96)(48,99,51,91)(55,118)(56,119)(57,120)(58,115)(59,116)(60,117) );
 
G=PermutationGroup([(1,86),(2,87),(3,88),(4,89),(5,90),(6,85),(7,77),(8,78),(9,73),(10,74),(11,75),(12,76),(13,83),(14,84),(15,79),(16,80),(17,81),(18,82),(19,63),(20,64),(21,65),(22,66),(23,61),(24,62),(25,69),(26,70),(27,71),(28,72),(29,67),(30,68),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120)], [(1,59),(2,60),(3,55),(4,56),(5,57),(6,58),(7,104),(8,105),(9,106),(10,107),(11,108),(12,103),(13,110),(14,111),(15,112),(16,113),(17,114),(18,109),(19,96),(20,91),(21,92),(22,93),(23,94),(24,95),(25,102),(26,97),(27,98),(28,99),(29,100),(30,101),(31,64),(32,65),(33,66),(34,61),(35,62),(36,63),(37,70),(38,71),(39,72),(40,67),(41,68),(42,69),(43,76),(44,77),(45,78),(46,73),(47,74),(48,75),(49,82),(50,83),(51,84),(52,79),(53,80),(54,81),(85,118),(86,119),(87,120),(88,115),(89,116),(90,117)], [(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),(91,92,93,94,95,96),(97,98,99,100,101,102),(103,104,105,106,107,108),(109,110,111,112,113,114),(115,116,117,118,119,120)], [(1,38,53,44,36),(2,39,54,45,31),(3,40,49,46,32),(4,41,50,47,33),(5,42,51,48,34),(6,37,52,43,35),(7,19,119,27,16),(8,20,120,28,17),(9,21,115,29,18),(10,22,116,30,13),(11,23,117,25,14),(12,24,118,26,15),(55,67,82,73,65),(56,68,83,74,66),(57,69,84,75,61),(58,70,79,76,62),(59,71,80,77,63),(60,72,81,78,64),(85,97,112,103,95),(86,98,113,104,96),(87,99,114,105,91),(88,100,109,106,92),(89,101,110,107,93),(90,102,111,108,94)], [(1,89),(2,90),(3,85),(4,86),(5,87),(6,88),(7,68,16,66),(8,69,17,61),(9,70,18,62),(10,71,13,63),(11,72,14,64),(12,67,15,65),(19,74,27,83),(20,75,28,84),(21,76,29,79),(22,77,30,80),(23,78,25,81),(24,73,26,82),(31,108,39,111),(32,103,40,112),(33,104,41,113),(34,105,42,114),(35,106,37,109),(36,107,38,110),(43,100,52,92),(44,101,53,93),(45,102,54,94),(46,97,49,95),(47,98,50,96),(48,99,51,91),(55,118),(56,119),(57,120),(58,115),(59,116),(60,117)])
 

Matrix representation ►G ⊆ GL6(𝔽61)

100000
010000
0060000
0006000
0000600
0000060
,
6000000
0600000
0060000
0006000
0000600
0000060
,
1400000
010000
0014000
0001400
0000140
0000014
,
100000
010000
0060606060
001000
000100
000010
,
5000000
0500000
000010
001000
000001
000100

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],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[14,0,0,0,0,0,0,1,0,0,0,0,0,0,14,0,0,0,0,0,0,14,0,0,0,0,0,0,14,0,0,0,0,0,0,14],[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],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;
 

120 conjugacy classes

class 1 2A···2G2H···2O3A3B4A···4P 5 6A···6N6O···6AD10A···10G12A···12AF15A15B30A···30N
order12···22···2334···456···66···610···1012···12151530···30
size11···15···5115···541···15···54···45···5444···4

120 irreducible representations

dim11111111114444
type+++++
imageC1C2C2C3C4C4C6C6C12C12F5C2×F5C3×F5C6×F5
kernelF5×C22×C6C2×C6×F5D5×C22×C6C23×F5D5×C2×C6C22×C30C22×F5C23×D5C22×D5C22×C10C22×C6C2×C6C23C22
# reps1141214228228417214

In GAP, Magma, Sage, TeX 

F_5\times C_2^2\times C_6
 
% in TeX
 
G:=Group("F5xC2^2xC6");
 
// GroupNames label
 
G:=SmallGroup(480,1205);
 
// by ID
 
G=gap.SmallGroup(480,1205);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,336,9414,433]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^6=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
 
// generators/relations
 

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