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## G = F5×C22×C6order 480 = 25·3·5

### Direct product of C22×C6 and F5

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
 Upper central C1 — C22×C6

Generators and relations for F5×C22×C6
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 >

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

Smallest permutation representation of F5×C22×C6
On 120 points
Generators in S120
(1 86)(2 87)(3 88)(4 89)(5 90)(6 85)(7 82)(8 83)(9 84)(10 79)(11 80)(12 81)(13 78)(14 73)(15 74)(16 75)(17 76)(18 77)(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)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 105)(14 106)(15 107)(16 108)(17 103)(18 104)(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 43 37 49 36)(2 44 38 50 31)(3 45 39 51 32)(4 46 40 52 33)(5 47 41 53 34)(6 48 42 54 35)(7 19 119 17 26)(8 20 120 18 27)(9 21 115 13 28)(10 22 116 14 29)(11 23 117 15 30)(12 24 118 16 25)(55 78 72 84 65)(56 73 67 79 66)(57 74 68 80 61)(58 75 69 81 62)(59 76 70 82 63)(60 77 71 83 64)(85 108 102 114 95)(86 103 97 109 96)(87 104 98 110 91)(88 105 99 111 92)(89 106 100 112 93)(90 107 101 113 94)
(1 89)(2 90)(3 85)(4 86)(5 87)(6 88)(7 73 26 66)(8 74 27 61)(9 75 28 62)(10 76 29 63)(11 77 30 64)(12 78 25 65)(13 69 21 81)(14 70 22 82)(15 71 23 83)(16 72 24 84)(17 67 19 79)(18 68 20 80)(31 113 44 101)(32 114 45 102)(33 109 46 97)(34 110 47 98)(35 111 48 99)(36 112 43 100)(37 93 49 106)(38 94 50 107)(39 95 51 108)(40 96 52 103)(41 91 53 104)(42 92 54 105)(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)(7,82)(8,83)(9,84)(10,79)(11,80)(12,81)(13,78)(14,73)(15,74)(16,75)(17,76)(18,77)(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)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,105)(14,106)(15,107)(16,108)(17,103)(18,104)(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), (1,43,37,49,36)(2,44,38,50,31)(3,45,39,51,32)(4,46,40,52,33)(5,47,41,53,34)(6,48,42,54,35)(7,19,119,17,26)(8,20,120,18,27)(9,21,115,13,28)(10,22,116,14,29)(11,23,117,15,30)(12,24,118,16,25)(55,78,72,84,65)(56,73,67,79,66)(57,74,68,80,61)(58,75,69,81,62)(59,76,70,82,63)(60,77,71,83,64)(85,108,102,114,95)(86,103,97,109,96)(87,104,98,110,91)(88,105,99,111,92)(89,106,100,112,93)(90,107,101,113,94), (1,89)(2,90)(3,85)(4,86)(5,87)(6,88)(7,73,26,66)(8,74,27,61)(9,75,28,62)(10,76,29,63)(11,77,30,64)(12,78,25,65)(13,69,21,81)(14,70,22,82)(15,71,23,83)(16,72,24,84)(17,67,19,79)(18,68,20,80)(31,113,44,101)(32,114,45,102)(33,109,46,97)(34,110,47,98)(35,111,48,99)(36,112,43,100)(37,93,49,106)(38,94,50,107)(39,95,51,108)(40,96,52,103)(41,91,53,104)(42,92,54,105)(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)(7,82)(8,83)(9,84)(10,79)(11,80)(12,81)(13,78)(14,73)(15,74)(16,75)(17,76)(18,77)(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)(7,109)(8,110)(9,111)(10,112)(11,113)(12,114)(13,105)(14,106)(15,107)(16,108)(17,103)(18,104)(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), (1,43,37,49,36)(2,44,38,50,31)(3,45,39,51,32)(4,46,40,52,33)(5,47,41,53,34)(6,48,42,54,35)(7,19,119,17,26)(8,20,120,18,27)(9,21,115,13,28)(10,22,116,14,29)(11,23,117,15,30)(12,24,118,16,25)(55,78,72,84,65)(56,73,67,79,66)(57,74,68,80,61)(58,75,69,81,62)(59,76,70,82,63)(60,77,71,83,64)(85,108,102,114,95)(86,103,97,109,96)(87,104,98,110,91)(88,105,99,111,92)(89,106,100,112,93)(90,107,101,113,94), (1,89)(2,90)(3,85)(4,86)(5,87)(6,88)(7,73,26,66)(8,74,27,61)(9,75,28,62)(10,76,29,63)(11,77,30,64)(12,78,25,65)(13,69,21,81)(14,70,22,82)(15,71,23,83)(16,72,24,84)(17,67,19,79)(18,68,20,80)(31,113,44,101)(32,114,45,102)(33,109,46,97)(34,110,47,98)(35,111,48,99)(36,112,43,100)(37,93,49,106)(38,94,50,107)(39,95,51,108)(40,96,52,103)(41,91,53,104)(42,92,54,105)(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),(7,82),(8,83),(9,84),(10,79),(11,80),(12,81),(13,78),(14,73),(15,74),(16,75),(17,76),(18,77),(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),(7,109),(8,110),(9,111),(10,112),(11,113),(12,114),(13,105),(14,106),(15,107),(16,108),(17,103),(18,104),(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)], [(1,43,37,49,36),(2,44,38,50,31),(3,45,39,51,32),(4,46,40,52,33),(5,47,41,53,34),(6,48,42,54,35),(7,19,119,17,26),(8,20,120,18,27),(9,21,115,13,28),(10,22,116,14,29),(11,23,117,15,30),(12,24,118,16,25),(55,78,72,84,65),(56,73,67,79,66),(57,74,68,80,61),(58,75,69,81,62),(59,76,70,82,63),(60,77,71,83,64),(85,108,102,114,95),(86,103,97,109,96),(87,104,98,110,91),(88,105,99,111,92),(89,106,100,112,93),(90,107,101,113,94)], [(1,89),(2,90),(3,85),(4,86),(5,87),(6,88),(7,73,26,66),(8,74,27,61),(9,75,28,62),(10,76,29,63),(11,77,30,64),(12,78,25,65),(13,69,21,81),(14,70,22,82),(15,71,23,83),(16,72,24,84),(17,67,19,79),(18,68,20,80),(31,113,44,101),(32,114,45,102),(33,109,46,97),(34,110,47,98),(35,111,48,99),(36,112,43,100),(37,93,49,106),(38,94,50,107),(39,95,51,108),(40,96,52,103),(41,91,53,104),(42,92,54,105),(55,118),(56,119),(57,120),(58,115),(59,116),(60,117)]])

120 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 4A ··· 4P 5 6A ··· 6N 6O ··· 6AD 10A ··· 10G 12A ··· 12AF 15A 15B 30A ··· 30N order 1 2 ··· 2 2 ··· 2 3 3 4 ··· 4 5 6 ··· 6 6 ··· 6 10 ··· 10 12 ··· 12 15 15 30 ··· 30 size 1 1 ··· 1 5 ··· 5 1 1 5 ··· 5 4 1 ··· 1 5 ··· 5 4 ··· 4 5 ··· 5 4 4 4 ··· 4

120 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 F5×C22×C6 C2×C6×F5 D5×C22×C6 C23×F5 D5×C2×C6 C22×C30 C22×F5 C23×D5 C22×D5 C22×C10 C22×C6 C2×C6 C23 C22 # reps 1 14 1 2 14 2 28 2 28 4 1 7 2 14

Matrix representation of F5×C22×C6 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
,
 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 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 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

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] >;

F5×C22×C6 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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