direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C2×C20, C60⋊13C22, C30.49C23, C30⋊9(C2×C4), C6⋊1(C2×C20), C12⋊3(C2×C10), (C2×C60)⋊13C2, (C2×C12)⋊5C10, C3⋊1(C22×C20), C15⋊10(C22×C4), D6.4(C2×C10), (C2×C10).37D6, (C2×Dic3)⋊5C10, Dic3⋊3(C2×C10), C6.2(C22×C10), C22.9(S3×C10), (C10×Dic3)⋊11C2, (C22×S3).2C10, (C2×C30).48C22, C10.39(C22×S3), (S3×C10).15C22, (C5×Dic3)⋊10C22, C2.1(S3×C2×C10), (S3×C2×C10).4C2, (C2×C6).9(C2×C10), (C2×C20)○(C5×Dic3), (C2×C20)○(C10×Dic3), SmallGroup(240,166)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C2×C20 |
Generators and relations for S3×C2×C20
G = < a,b,c,d | a2=b20=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 184 in 108 conjugacy classes, 70 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C23, C10, C10, C10, Dic3, C12, D6, C2×C6, C15, C22×C4, C20, C20, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C30, C2×C20, C2×C20, C22×C10, S3×C2×C4, C5×Dic3, C60, S3×C10, C2×C30, C22×C20, S3×C20, C10×Dic3, C2×C60, S3×C2×C10, S3×C2×C20
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C23, C10, D6, C22×C4, C20, C2×C10, C4×S3, C22×S3, C5×S3, C2×C20, C22×C10, S3×C2×C4, S3×C10, C22×C20, S3×C20, S3×C2×C10, S3×C2×C20
►On 120 points
Generators in S120
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 41)(18 42)(19 43)(20 44)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(33 119)(34 120)(35 101)(36 102)(37 103)(38 104)(39 105)(40 106)(61 100)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)
(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 71 109)(2 72 110)(3 73 111)(4 74 112)(5 75 113)(6 76 114)(7 77 115)(8 78 116)(9 79 117)(10 80 118)(11 61 119)(12 62 120)(13 63 101)(14 64 102)(15 65 103)(16 66 104)(17 67 105)(18 68 106)(19 69 107)(20 70 108)(21 43 88)(22 44 89)(23 45 90)(24 46 91)(25 47 92)(26 48 93)(27 49 94)(28 50 95)(29 51 96)(30 52 97)(31 53 98)(32 54 99)(33 55 100)(34 56 81)(35 57 82)(36 58 83)(37 59 84)(38 60 85)(39 41 86)(40 42 87)
(21 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(61 119)(62 120)(63 101)(64 102)(65 103)(66 104)(67 105)(68 106)(69 107)(70 108)(71 109)(72 110)(73 111)(74 112)(75 113)(76 114)(77 115)(78 116)(79 117)(80 118)
G:=sub<Sym(120)| (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,41)(18,42)(19,43)(20,44)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,101)(36,102)(37,103)(38,104)(39,105)(40,106)(61,100)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,99), (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), (1,71,109)(2,72,110)(3,73,111)(4,74,112)(5,75,113)(6,76,114)(7,77,115)(8,78,116)(9,79,117)(10,80,118)(11,61,119)(12,62,120)(13,63,101)(14,64,102)(15,65,103)(16,66,104)(17,67,105)(18,68,106)(19,69,107)(20,70,108)(21,43,88)(22,44,89)(23,45,90)(24,46,91)(25,47,92)(26,48,93)(27,49,94)(28,50,95)(29,51,96)(30,52,97)(31,53,98)(32,54,99)(33,55,100)(34,56,81)(35,57,82)(36,58,83)(37,59,84)(38,60,85)(39,41,86)(40,42,87), (21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(61,119)(62,120)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118)>;
G:=Group( (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,41)(18,42)(19,43)(20,44)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,101)(36,102)(37,103)(38,104)(39,105)(40,106)(61,100)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,99), (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), (1,71,109)(2,72,110)(3,73,111)(4,74,112)(5,75,113)(6,76,114)(7,77,115)(8,78,116)(9,79,117)(10,80,118)(11,61,119)(12,62,120)(13,63,101)(14,64,102)(15,65,103)(16,66,104)(17,67,105)(18,68,106)(19,69,107)(20,70,108)(21,43,88)(22,44,89)(23,45,90)(24,46,91)(25,47,92)(26,48,93)(27,49,94)(28,50,95)(29,51,96)(30,52,97)(31,53,98)(32,54,99)(33,55,100)(34,56,81)(35,57,82)(36,58,83)(37,59,84)(38,60,85)(39,41,86)(40,42,87), (21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(61,119)(62,120)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118) );
G=PermutationGroup([[(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,41),(18,42),(19,43),(20,44),(21,107),(22,108),(23,109),(24,110),(25,111),(26,112),(27,113),(28,114),(29,115),(30,116),(31,117),(32,118),(33,119),(34,120),(35,101),(36,102),(37,103),(38,104),(39,105),(40,106),(61,100),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,91),(73,92),(74,93),(75,94),(76,95),(77,96),(78,97),(79,98),(80,99)], [(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)], [(1,71,109),(2,72,110),(3,73,111),(4,74,112),(5,75,113),(6,76,114),(7,77,115),(8,78,116),(9,79,117),(10,80,118),(11,61,119),(12,62,120),(13,63,101),(14,64,102),(15,65,103),(16,66,104),(17,67,105),(18,68,106),(19,69,107),(20,70,108),(21,43,88),(22,44,89),(23,45,90),(24,46,91),(25,47,92),(26,48,93),(27,49,94),(28,50,95),(29,51,96),(30,52,97),(31,53,98),(32,54,99),(33,55,100),(34,56,81),(35,57,82),(36,58,83),(37,59,84),(38,60,85),(39,41,86),(40,42,87)], [(21,88),(22,89),(23,90),(24,91),(25,92),(26,93),(27,94),(28,95),(29,96),(30,97),(31,98),(32,99),(33,100),(34,81),(35,82),(36,83),(37,84),(38,85),(39,86),(40,87),(61,119),(62,120),(63,101),(64,102),(65,103),(66,104),(67,105),(68,106),(69,107),(70,108),(71,109),(72,110),(73,111),(74,112),(75,113),(76,114),(77,115),(78,116),(79,117),(80,118)]])
S3×C2×C20 is a maximal subgroup of
C60.94D4 (S3×C20)⋊5C4 D6⋊Dic5⋊C2 D6⋊Dic10 C60.45D4 C60.46D4 (S3×C20)⋊7C4 C15⋊17(C4×D4) C15⋊22(C4×D4) D10⋊C4⋊S3 D6⋊D20 C60⋊4D4 C60⋊6D4
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 10M | ··· | 10AB | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20AF | 30A | ··· | 30L | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
120 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 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | S3 | D6 | D6 | C4×S3 | C5×S3 | S3×C10 | S3×C10 | S3×C20 |
| kernel | S3×C2×C20 | S3×C20 | C10×Dic3 | C2×C60 | S3×C2×C10 | S3×C10 | S3×C2×C4 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | D6 | C2×C20 | C20 | C2×C10 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 16 | 4 | 4 | 4 | 32 | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 |
Matrix representation of S3×C2×C20 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 9 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 1 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[9,0,0,0,0,50,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,60],[60,0,0,0,0,60,0,0,0,0,0,1,0,0,1,0] >;
S3×C2×C20 in GAP, Magma, Sage, TeX
S_3\times C_2\times C_{20} % in TeX
G:=Group("S3xC2xC20"); // GroupNames label
G:=SmallGroup(240,166);
// by ID
G=gap.SmallGroup(240,166);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-3,194,5765]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=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