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G = C3×D60order 360 = 23·32·5

Direct product of C3 and D60

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C3×D60
 Chief series C1 — C5 — C15 — C30 — C3×C30 — C6×D15 — C3×D60
 Lower central C15 — C30 — C3×D60
 Upper central C1 — C6 — C12

Generators and relations for C3×D60
G = < a,b,c | a3=b60=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 372 in 70 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, D4, C32, D5, C10, C12, C12, D6, C2×C6, C15, C15, C3×S3, C3×C6, C20, D10, D12, C3×D4, C3×D5, D15, C30, C30, C3×C12, S3×C6, D20, C3×C15, C60, C60, C6×D5, D30, C3×D12, C3×D15, C3×C30, C3×D20, D60, C3×C60, C6×D15, C3×D60
Quotients: C1, C2, C3, C22, S3, C6, D4, D5, D6, C2×C6, C3×S3, D10, D12, C3×D4, C3×D5, D15, S3×C6, D20, C6×D5, D30, C3×D12, C3×D15, C3×D20, D60, C6×D15, C3×D60

Smallest permutation representation of C3×D60
On 120 points
Generators in S120
(1 41 21)(2 42 22)(3 43 23)(4 44 24)(5 45 25)(6 46 26)(7 47 27)(8 48 28)(9 49 29)(10 50 30)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)(61 81 101)(62 82 102)(63 83 103)(64 84 104)(65 85 105)(66 86 106)(67 87 107)(68 88 108)(69 89 109)(70 90 110)(71 91 111)(72 92 112)(73 93 113)(74 94 114)(75 95 115)(76 96 116)(77 97 117)(78 98 118)(79 99 119)(80 100 120)
(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 75)(2 74)(3 73)(4 72)(5 71)(6 70)(7 69)(8 68)(9 67)(10 66)(11 65)(12 64)(13 63)(14 62)(15 61)(16 120)(17 119)(18 118)(19 117)(20 116)(21 115)(22 114)(23 113)(24 112)(25 111)(26 110)(27 109)(28 108)(29 107)(30 106)(31 105)(32 104)(33 103)(34 102)(35 101)(36 100)(37 99)(38 98)(39 97)(40 96)(41 95)(42 94)(43 93)(44 92)(45 91)(46 90)(47 89)(48 88)(49 87)(50 86)(51 85)(52 84)(53 83)(54 82)(55 81)(56 80)(57 79)(58 78)(59 77)(60 76)

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

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

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

99 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4 5A 5B 6A 6B 6C 6D 6E 6F 6G 6H 6I 10A 10B 12A ··· 12H 15A ··· 15P 20A 20B 20C 20D 30A ··· 30P 60A ··· 60AF order 1 2 2 2 3 3 3 3 3 4 5 5 6 6 6 6 6 6 6 6 6 10 10 12 ··· 12 15 ··· 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 30 30 1 1 2 2 2 2 2 2 1 1 2 2 2 30 30 30 30 2 2 2 ··· 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2

99 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D4 D5 D6 C3×S3 D10 D12 C3×D4 C3×D5 D15 S3×C6 D20 C6×D5 D30 C3×D12 C3×D15 C3×D20 D60 C6×D15 C3×D60 kernel C3×D60 C3×C60 C6×D15 D60 C60 D30 C60 C3×C15 C3×C12 C30 C20 C3×C6 C15 C15 C12 C12 C10 C32 C6 C6 C5 C4 C3 C3 C2 C1 # reps 1 1 2 2 2 4 1 1 2 1 2 2 2 2 4 4 2 4 4 4 4 8 8 8 8 16

Matrix representation of C3×D60 in GL2(𝔽61) generated by

 47 0 0 47
,
 44 0 0 43
,
 0 43 44 0
G:=sub<GL(2,GF(61))| [47,0,0,47],[44,0,0,43],[0,44,43,0] >;

C3×D60 in GAP, Magma, Sage, TeX

C_3\times D_{60}
% in TeX

G:=Group("C3xD60");
// GroupNames label

G:=SmallGroup(360,102);
// by ID

G=gap.SmallGroup(360,102);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,169,79,1444,10373]);
// Polycyclic

G:=Group<a,b,c|a^3=b^60=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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