direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C2×C30, C62⋊4C10, C6⋊(C2×C30), C30⋊4(C2×C6), (C2×C6)⋊5C30, C3⋊(C22×C30), (C2×C30)⋊11C6, (C6×C30)⋊10C2, C15⋊4(C22×C6), (C3×C30)⋊9C22, (C3×C15)⋊10C23, C32⋊2(C22×C10), (C3×C6)⋊2(C2×C10), SmallGroup(360,158)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C2×C30 |
Generators and relations for S3×C2×C30
G = < a,b,c,d | a2=b30=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 236 in 138 conjugacy classes, 84 normal (20 characteristic)
C1, C2, C2, C3, C3, C22, C22, C5, S3, C6, C6, C23, C32, C10, C10, D6, C2×C6, C2×C6, C15, C15, C3×S3, C3×C6, C2×C10, C2×C10, C22×S3, C22×C6, C5×S3, C30, C30, S3×C6, C62, C22×C10, C3×C15, S3×C10, C2×C30, C2×C30, S3×C2×C6, S3×C15, C3×C30, S3×C2×C10, C22×C30, S3×C30, C6×C30, S3×C2×C30
Quotients: C1, C2, C3, C22, C5, S3, C6, C23, C10, D6, C2×C6, C15, C3×S3, C2×C10, C22×S3, C22×C6, C5×S3, C30, S3×C6, C22×C10, S3×C10, C2×C30, S3×C2×C6, S3×C15, S3×C2×C10, C22×C30, S3×C30, S3×C2×C30
►On 120 points
Generators in S120
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(61 98)(62 99)(63 100)(64 101)(65 102)(66 103)(67 104)(68 105)(69 106)(70 107)(71 108)(72 109)(73 110)(74 111)(75 112)(76 113)(77 114)(78 115)(79 116)(80 117)(81 118)(82 119)(83 120)(84 91)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)
(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 21 11)(2 22 12)(3 23 13)(4 24 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)(61 71 81)(62 72 82)(63 73 83)(64 74 84)(65 75 85)(66 76 86)(67 77 87)(68 78 88)(69 79 89)(70 80 90)(91 101 111)(92 102 112)(93 103 113)(94 104 114)(95 105 115)(96 106 116)(97 107 117)(98 108 118)(99 109 119)(100 110 120)
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(21 90)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 116)(32 117)(33 118)(34 119)(35 120)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)
G:=sub<Sym(120)| (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(61,98)(62,99)(63,100)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,91)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97), (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), (1,21,11)(2,22,12)(3,23,13)(4,24,14)(5,25,15)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50)(61,71,81)(62,72,82)(63,73,83)(64,74,84)(65,75,85)(66,76,86)(67,77,87)(68,78,88)(69,79,89)(70,80,90)(91,101,111)(92,102,112)(93,103,113)(94,104,114)(95,105,115)(96,106,116)(97,107,117)(98,108,118)(99,109,119)(100,110,120), (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,116)(32,117)(33,118)(34,119)(35,120)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115)>;
G:=Group( (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(61,98)(62,99)(63,100)(64,101)(65,102)(66,103)(67,104)(68,105)(69,106)(70,107)(71,108)(72,109)(73,110)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,91)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97), (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), (1,21,11)(2,22,12)(3,23,13)(4,24,14)(5,25,15)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50)(61,71,81)(62,72,82)(63,73,83)(64,74,84)(65,75,85)(66,76,86)(67,77,87)(68,78,88)(69,79,89)(70,80,90)(91,101,111)(92,102,112)(93,103,113)(94,104,114)(95,105,115)(96,106,116)(97,107,117)(98,108,118)(99,109,119)(100,110,120), (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,61)(23,62)(24,63)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,116)(32,117)(33,118)(34,119)(35,120)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113)(59,114)(60,115) );
G=PermutationGroup([[(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,50),(30,51),(61,98),(62,99),(63,100),(64,101),(65,102),(66,103),(67,104),(68,105),(69,106),(70,107),(71,108),(72,109),(73,110),(74,111),(75,112),(76,113),(77,114),(78,115),(79,116),(80,117),(81,118),(82,119),(83,120),(84,91),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97)], [(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)], [(1,21,11),(2,22,12),(3,23,13),(4,24,14),(5,25,15),(6,26,16),(7,27,17),(8,28,18),(9,29,19),(10,30,20),(31,51,41),(32,52,42),(33,53,43),(34,54,44),(35,55,45),(36,56,46),(37,57,47),(38,58,48),(39,59,49),(40,60,50),(61,71,81),(62,72,82),(63,73,83),(64,74,84),(65,75,85),(66,76,86),(67,77,87),(68,78,88),(69,79,89),(70,80,90),(91,101,111),(92,102,112),(93,103,113),(94,104,114),(95,105,115),(96,106,116),(97,107,117),(98,108,118),(99,109,119),(100,110,120)], [(1,70),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,77),(9,78),(10,79),(11,80),(12,81),(13,82),(14,83),(15,84),(16,85),(17,86),(18,87),(19,88),(20,89),(21,90),(22,61),(23,62),(24,63),(25,64),(26,65),(27,66),(28,67),(29,68),(30,69),(31,116),(32,117),(33,118),(34,119),(35,120),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111),(57,112),(58,113),(59,114),(60,115)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6W | 10A | ··· | 10L | 10M | ··· | 10AB | 15A | ··· | 15H | 15I | ··· | 15T | 30A | ··· | 30X | 30Y | ··· | 30BH | 30BI | ··· | 30CN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | ··· | 15 | 15 | ··· | 15 | 30 | ··· | 30 | 30 | ··· | 30 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C5 | C6 | C6 | C10 | C10 | C15 | C30 | C30 | S3 | D6 | C3×S3 | C5×S3 | S3×C6 | S3×C10 | S3×C15 | S3×C30 |
| kernel | S3×C2×C30 | S3×C30 | C6×C30 | S3×C2×C10 | S3×C2×C6 | S3×C10 | C2×C30 | S3×C6 | C62 | C22×S3 | D6 | C2×C6 | C2×C30 | C30 | C2×C10 | C2×C6 | C10 | C6 | C22 | C2 |
| # reps | 1 | 6 | 1 | 2 | 4 | 12 | 2 | 24 | 4 | 8 | 48 | 8 | 1 | 3 | 2 | 4 | 6 | 12 | 8 | 24 |
Matrix representation of S3×C2×C30 ►in GL4(𝔽31) generated by
| 1 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 22 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 25 |
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(31))| [1,0,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[22,0,0,0,0,5,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,5,0,0,0,0,25],[30,0,0,0,0,30,0,0,0,0,0,1,0,0,1,0] >;
S3×C2×C30 in GAP, Magma, Sage, TeX
S_3\times C_2\times C_{30} % in TeX
G:=Group("S3xC2xC30"); // GroupNames label
G:=SmallGroup(360,158);
// by ID
G=gap.SmallGroup(360,158);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-5,-3,8645]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^30=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations