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## G = C2×Dic3×F5order 480 = 25·3·5

### Direct product of C2, Dic3 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C2×Dic3×F5
 Chief series C1 — C5 — C15 — C3×D5 — C6×D5 — C6×F5 — Dic3×F5 — C2×Dic3×F5
 Lower central C15 — C2×Dic3×F5
 Upper central C1 — C22

Generators and relations for C2×Dic3×F5
G = < a,b,c,d,e | a2=b6=d5=e4=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 884 in 216 conjugacy classes, 94 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, C6, C6, C6, C2×C4, C23, D5, D5, C10, C10, Dic3, Dic3, C12, C2×C6, C2×C6, C15, C42, C22×C4, Dic5, C20, F5, F5, D10, D10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C3×D5, C3×D5, C30, C30, C2×C42, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C4×Dic3, C22×Dic3, C22×C12, C5×Dic3, Dic15, C3×F5, C3⋊F5, C6×D5, C6×D5, C2×C30, C4×F5, C2×C4×D5, C22×F5, C22×F5, C2×C4×Dic3, D5×Dic3, C10×Dic3, C2×Dic15, C6×F5, C2×C3⋊F5, D5×C2×C6, C2×C4×F5, Dic3×F5, C2×D5×Dic3, C2×C6×F5, C22×C3⋊F5, C2×Dic3×F5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C42, C22×C4, F5, C4×S3, C2×Dic3, C22×S3, C2×C42, C2×F5, C4×Dic3, S3×C2×C4, C22×Dic3, C4×F5, C22×F5, C2×C4×Dic3, S3×F5, C2×C4×F5, Dic3×F5, C2×S3×F5, C2×Dic3×F5

Smallest permutation representation of C2×Dic3×F5
On 120 points
Generators in S120
(1 58)(2 59)(3 60)(4 55)(5 56)(6 57)(7 112)(8 113)(9 114)(10 109)(11 110)(12 111)(13 108)(14 103)(15 104)(16 105)(17 106)(18 107)(19 98)(20 99)(21 100)(22 101)(23 102)(24 97)(25 94)(26 95)(27 96)(28 91)(29 92)(30 93)(31 72)(32 67)(33 68)(34 69)(35 70)(36 71)(37 62)(38 63)(39 64)(40 65)(41 66)(42 61)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)
(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 89 4 86)(2 88 5 85)(3 87 6 90)(7 84 10 81)(8 83 11 80)(9 82 12 79)(13 78 16 75)(14 77 17 74)(15 76 18 73)(19 72 22 69)(20 71 23 68)(21 70 24 67)(25 66 28 63)(26 65 29 62)(27 64 30 61)(31 101 34 98)(32 100 35 97)(33 99 36 102)(37 95 40 92)(38 94 41 91)(39 93 42 96)(43 104 46 107)(44 103 47 106)(45 108 48 105)(49 114 52 111)(50 113 53 110)(51 112 54 109)(55 116 58 119)(56 115 59 118)(57 120 60 117)
(1 47 32 51 39)(2 48 33 52 40)(3 43 34 53 41)(4 44 35 54 42)(5 45 36 49 37)(6 46 31 50 38)(7 30 119 17 21)(8 25 120 18 22)(9 26 115 13 23)(10 27 116 14 24)(11 28 117 15 19)(12 29 118 16 20)(55 74 70 84 61)(56 75 71 79 62)(57 76 72 80 63)(58 77 67 81 64)(59 78 68 82 65)(60 73 69 83 66)(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)
(1 119 4 116)(2 120 5 117)(3 115 6 118)(7 44 24 39)(8 45 19 40)(9 46 20 41)(10 47 21 42)(11 48 22 37)(12 43 23 38)(13 31 29 53)(14 32 30 54)(15 33 25 49)(16 34 26 50)(17 35 27 51)(18 36 28 52)(55 86 58 89)(56 87 59 90)(57 88 60 85)(61 109 77 100)(62 110 78 101)(63 111 73 102)(64 112 74 97)(65 113 75 98)(66 114 76 99)(67 93 84 103)(68 94 79 104)(69 95 80 105)(70 96 81 106)(71 91 82 107)(72 92 83 108)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E ··· 4L 4M ··· 4X 5 6A 6B 6C 6D 6E 6F 6G 10A 10B 10C 12A ··· 12H 15 20A 20B 20C 20D 30A 30B 30C order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 ··· 4 4 ··· 4 5 6 6 6 6 6 6 6 10 10 10 12 ··· 12 15 20 20 20 20 30 30 30 size 1 1 1 1 5 5 5 5 2 3 3 3 3 5 ··· 5 15 ··· 15 4 2 2 2 10 10 10 10 4 4 4 10 ··· 10 8 12 12 12 12 8 8 8

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 8 8 8 type + + + + + + - + + + + + + - + image C1 C2 C2 C2 C2 C4 C4 C4 C4 C4 S3 Dic3 D6 D6 C4×S3 C4×S3 F5 C2×F5 C2×F5 C4×F5 S3×F5 Dic3×F5 C2×S3×F5 kernel C2×Dic3×F5 Dic3×F5 C2×D5×Dic3 C2×C6×F5 C22×C3⋊F5 D5×Dic3 C10×Dic3 C2×Dic15 C6×F5 C2×C3⋊F5 C22×F5 C2×F5 C2×F5 C22×D5 D10 C2×C10 C2×Dic3 Dic3 C2×C6 C6 C22 C2 C2 # reps 1 4 1 1 1 4 2 2 8 8 1 4 2 1 6 2 1 2 1 4 1 2 1

Matrix representation of C2×Dic3×F5 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
,
 0 60 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 11 0 0 0 0 0 50 50 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
,
 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 60 60 60 60

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],[0,1,0,0,0,0,60,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,50,0,0,0,0,0,50,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,60,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,1,0,60] >;

C2×Dic3×F5 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3\times F_5
% in TeX

G:=Group("C2xDic3xF5");
// GroupNames label

G:=SmallGroup(480,998);
// by ID

G=gap.SmallGroup(480,998);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,1356,9414,2379]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=d^5=e^4=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,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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