direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C40, C24⋊4C10, C120⋊12C2, D6.2C20, C20.56D6, C60.73C22, Dic3.2C20, C3⋊C8⋊6C10, C3⋊1(C2×C40), C15⋊12(C2×C8), C2.1(S3×C20), C6.1(C2×C20), C40○(C5×Dic3), (C4×S3).3C10, (S3×C20).6C2, (S3×C10).6C4, C10.22(C4×S3), C4.12(S3×C10), C30.45(C2×C4), C12.12(C2×C10), (C5×Dic3).6C4, C40○(C5×C3⋊C8), (C5×C3⋊C8)⋊13C2, SmallGroup(240,49)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C40 |
Generators and relations for S3×C40
G = < a,b,c | a40=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C40
►On 120 points
Generators in S120
(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 64 106)(2 65 107)(3 66 108)(4 67 109)(5 68 110)(6 69 111)(7 70 112)(8 71 113)(9 72 114)(10 73 115)(11 74 116)(12 75 117)(13 76 118)(14 77 119)(15 78 120)(16 79 81)(17 80 82)(18 41 83)(19 42 84)(20 43 85)(21 44 86)(22 45 87)(23 46 88)(24 47 89)(25 48 90)(26 49 91)(27 50 92)(28 51 93)(29 52 94)(30 53 95)(31 54 96)(32 55 97)(33 56 98)(34 57 99)(35 58 100)(36 59 101)(37 60 102)(38 61 103)(39 62 104)(40 63 105)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)(67 89)(68 90)(69 91)(70 92)(71 93)(72 94)(73 95)(74 96)(75 97)(76 98)(77 99)(78 100)(79 101)(80 102)
G:=sub<Sym(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), (1,64,106)(2,65,107)(3,66,108)(4,67,109)(5,68,110)(6,69,111)(7,70,112)(8,71,113)(9,72,114)(10,73,115)(11,74,116)(12,75,117)(13,76,118)(14,77,119)(15,78,120)(16,79,81)(17,80,82)(18,41,83)(19,42,84)(20,43,85)(21,44,86)(22,45,87)(23,46,88)(24,47,89)(25,48,90)(26,49,91)(27,50,92)(28,51,93)(29,52,94)(30,53,95)(31,54,96)(32,55,97)(33,56,98)(34,57,99)(35,58,100)(36,59,101)(37,60,102)(38,61,103)(39,62,104)(40,63,105), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94)(73,95)(74,96)(75,97)(76,98)(77,99)(78,100)(79,101)(80,102)>;
G:=Group( (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), (1,64,106)(2,65,107)(3,66,108)(4,67,109)(5,68,110)(6,69,111)(7,70,112)(8,71,113)(9,72,114)(10,73,115)(11,74,116)(12,75,117)(13,76,118)(14,77,119)(15,78,120)(16,79,81)(17,80,82)(18,41,83)(19,42,84)(20,43,85)(21,44,86)(22,45,87)(23,46,88)(24,47,89)(25,48,90)(26,49,91)(27,50,92)(28,51,93)(29,52,94)(30,53,95)(31,54,96)(32,55,97)(33,56,98)(34,57,99)(35,58,100)(36,59,101)(37,60,102)(38,61,103)(39,62,104)(40,63,105), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,103)(42,104)(43,105)(44,106)(45,107)(46,108)(47,109)(48,110)(49,111)(50,112)(51,113)(52,114)(53,115)(54,116)(55,117)(56,118)(57,119)(58,120)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94)(73,95)(74,96)(75,97)(76,98)(77,99)(78,100)(79,101)(80,102) );
G=PermutationGroup([[(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)], [(1,64,106),(2,65,107),(3,66,108),(4,67,109),(5,68,110),(6,69,111),(7,70,112),(8,71,113),(9,72,114),(10,73,115),(11,74,116),(12,75,117),(13,76,118),(14,77,119),(15,78,120),(16,79,81),(17,80,82),(18,41,83),(19,42,84),(20,43,85),(21,44,86),(22,45,87),(23,46,88),(24,47,89),(25,48,90),(26,49,91),(27,50,92),(28,51,93),(29,52,94),(30,53,95),(31,54,96),(32,55,97),(33,56,98),(34,57,99),(35,58,100),(36,59,101),(37,60,102),(38,61,103),(39,62,104),(40,63,105)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,103),(42,104),(43,105),(44,106),(45,107),(46,108),(47,109),(48,110),(49,111),(50,112),(51,113),(52,114),(53,115),(54,116),(55,117),(56,118),(57,119),(58,120),(59,81),(60,82),(61,83),(62,84),(63,85),(64,86),(65,87),(66,88),(67,89),(68,90),(69,91),(70,92),(71,93),(72,94),(73,95),(74,96),(75,97),(76,98),(77,99),(78,100),(79,101),(80,102)]])
S3×C40 is a maximal subgroup of
C40.52D6 C40.54D6 C40.55D6 D6.1D20 D40⋊7S3 D120⋊5C2
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40P | 40Q | ··· | 40AF | 60A | ··· | 60H | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C10 | C20 | C20 | C40 | S3 | D6 | C4×S3 | C5×S3 | S3×C8 | S3×C10 | S3×C20 | S3×C40 |
| kernel | S3×C40 | C5×C3⋊C8 | C120 | S3×C20 | C5×Dic3 | S3×C10 | S3×C8 | C5×S3 | C3⋊C8 | C24 | C4×S3 | Dic3 | D6 | S3 | C40 | C20 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 4 | 4 | 4 | 8 | 8 | 32 | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 16 |
Matrix representation of S3×C40 ►in GL2(𝔽41) generated by
| 6 | 0 |
| 0 | 6 |
| 40 | 34 |
| 6 | 0 |
| 40 | 34 |
| 0 | 1 |
G:=sub<GL(2,GF(41))| [6,0,0,6],[40,6,34,0],[40,0,34,1] >;
S3×C40 in GAP, Magma, Sage, TeX
S_3\times C_{40} % in TeX
G:=Group("S3xC40"); // GroupNames label
G:=SmallGroup(240,49);
// by ID
G=gap.SmallGroup(240,49);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-3,127,69,5765]);
// Polycyclic
G:=Group<a,b,c|a^40=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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