direct product, metabelian, supersoluble, monomial, A-group
Aliases: C5×S3×Dic3, C30.35D6, (C3×S3)⋊C20, D6.(C5×S3), C3⋊3(S3×C20), C10.13S32, (S3×C6).C10, (S3×C15)⋊5C4, C15⋊18(C4×S3), C6.1(S3×C10), C32⋊2(C2×C20), (S3×C30).3C2, (S3×C10).2S3, C3⋊Dic3⋊1C10, C15⋊9(C2×Dic3), C3⋊1(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
| C32 — C5×S3×Dic3 |
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
►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 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | 12B | 15A | ··· | 15H | 15I | 15J | 15K | 15L | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30H | 30I | 30J | 30K | 30L | 30M | ··· | 30T | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 15 | ··· | 15 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 4 | 3 | 3 | 9 | 9 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 6 | 6 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 6 | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 3 | ··· | 3 | 9 | ··· | 9 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | S3 | S3 | Dic3 | D6 | C4×S3 | C5×S3 | C5×S3 | C5×Dic3 | S3×C10 | S3×C20 | S32 | S3×Dic3 | C5×S32 | C5×S3×Dic3 |
| kernel | C5×S3×Dic3 | Dic3×C15 | C5×C3⋊Dic3 | S3×C30 | S3×C15 | S3×Dic3 | C3×Dic3 | C3⋊Dic3 | S3×C6 | C3×S3 | C5×Dic3 | S3×C10 | C5×S3 | C30 | C15 | Dic3 | D6 | S3 | C6 | C3 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of C5×S3×Dic3 ►in GL6(𝔽61)
| 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 | 1 | 0 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 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 | 60 |
| 0 | 0 | 0 | 0 | 1 | 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 | 0 |
| 0 | 0 | 0 | 0 | 60 | 60 |
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