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G = C5×S3×Dic3order 360 = 23·32·5

Direct product of C5, S3 and Dic3

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

Aliases: C5×S3×Dic3, C30.35D6, (C3×S3)⋊C20, D6.(C5×S3), C33(S3×C20), C10.13S32, (S3×C6).C10, (S3×C15)⋊5C4, C1518(C4×S3), C6.1(S3×C10), C322(C2×C20), (S3×C30).3C2, (S3×C10).2S3, C3⋊Dic31C10, C159(C2×Dic3), C31(C10×Dic3), (Dic3×C15)⋊8C2, (C3×Dic3)⋊2C10, (C3×C30).27C22, C2.1(C5×S32), (C3×C15)⋊24(C2×C4), (C3×C6).1(C2×C10), (C5×C3⋊Dic3)⋊6C2, SmallGroup(360,72)

Series: Derived Chief Lower central Upper central

C1C32 — C5×S3×Dic3
C1C3C32C3×C6C3×C30S3×C30 — C5×S3×Dic3
C32 — C5×S3×Dic3
C1C10

Generators and relations for C5×S3×Dic3
 G = < a,b,c,d,e | a5=b3=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 164 in 70 conjugacy classes, 36 normal (28 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, C2×C4, C32, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C15, C15, C3×S3, C3×C6, C20, C2×C10, C4×S3, C2×Dic3, C5×S3, C30, C30, C3×Dic3, C3⋊Dic3, S3×C6, C2×C20, C3×C15, C5×Dic3, C5×Dic3, C60, S3×C10, C2×C30, S3×Dic3, S3×C15, C3×C30, S3×C20, C10×Dic3, Dic3×C15, C5×C3⋊Dic3, S3×C30, C5×S3×Dic3
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C10, Dic3, D6, C20, C2×C10, C4×S3, C2×Dic3, C5×S3, S32, C2×C20, C5×Dic3, S3×C10, S3×Dic3, S3×C20, C10×Dic3, C5×S32, C5×S3×Dic3

Smallest permutation representation of C5×S3×Dic3
On 120 points
Generators in S120
(1 25 19 13 7)(2 26 20 14 8)(3 27 21 15 9)(4 28 22 16 10)(5 29 23 17 11)(6 30 24 18 12)(31 55 49 43 37)(32 56 50 44 38)(33 57 51 45 39)(34 58 52 46 40)(35 59 53 47 41)(36 60 54 48 42)(61 85 79 73 67)(62 86 80 74 68)(63 87 81 75 69)(64 88 82 76 70)(65 89 83 77 71)(66 90 84 78 72)(91 115 109 103 97)(92 116 110 104 98)(93 117 111 105 99)(94 118 112 106 100)(95 119 113 107 101)(96 120 114 108 102)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 57 59)(56 58 60)(61 63 65)(62 64 66)(67 69 71)(68 70 72)(73 75 77)(74 76 78)(79 81 83)(80 82 84)(85 87 89)(86 88 90)(91 95 93)(92 96 94)(97 101 99)(98 102 100)(103 107 105)(104 108 106)(109 113 111)(110 114 112)(115 119 117)(116 120 118)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(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 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 32 4 35)(2 31 5 34)(3 36 6 33)(7 38 10 41)(8 37 11 40)(9 42 12 39)(13 44 16 47)(14 43 17 46)(15 48 18 45)(19 50 22 53)(20 49 23 52)(21 54 24 51)(25 56 28 59)(26 55 29 58)(27 60 30 57)(61 92 64 95)(62 91 65 94)(63 96 66 93)(67 98 70 101)(68 97 71 100)(69 102 72 99)(73 104 76 107)(74 103 77 106)(75 108 78 105)(79 110 82 113)(80 109 83 112)(81 114 84 111)(85 116 88 119)(86 115 89 118)(87 120 90 117)

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

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

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

90 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D5A5B5C5D6A6B6C6D6E10A10B10C10D10E···10L12A12B15A···15H15I15J15K15L20A···20H20I···20P30A···30H30I30J30K30L30M···30T60A···60H
order122233344445555666661010101010···10121215···151515151520···2020···2030···303030303030···3060···60
size1133224339911112246611113···3662···244443···39···92···244446···66···6

90 irreducible representations

dim111111111122222222224444
type++++++-++-
imageC1C2C2C2C4C5C10C10C10C20S3S3Dic3D6C4×S3C5×S3C5×S3C5×Dic3S3×C10S3×C20S32S3×Dic3C5×S32C5×S3×Dic3
kernelC5×S3×Dic3Dic3×C15C5×C3⋊Dic3S3×C30S3×C15S3×Dic3C3×Dic3C3⋊Dic3S3×C6C3×S3C5×Dic3S3×C10C5×S3C30C15Dic3D6S3C6C3C10C5C2C1
# reps1111444441611222448881144

Matrix representation of C5×S3×Dic3 in GL6(𝔽61)

100000
010000
009000
000900
000010
000001
,
010000
60600000
000100
00606000
000010
000001
,
0600000
6000000
000100
001000
000010
000001
,
6000000
0600000
001000
000100
00006060
000010
,
1100000
0110000
0060000
0006000
000010
00006060

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,60,0,0,0,0,1,60,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,60] >;

C5×S3×Dic3 in GAP, Magma, Sage, TeX

C_5\times S_3\times {\rm Dic}_3
% in TeX

G:=Group("C5xS3xDic3");
// GroupNames label

G:=SmallGroup(360,72);
// by ID

G=gap.SmallGroup(360,72);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,-3,127,1210,8645]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^3=c^2=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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