direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D40, C15⋊5D8, C40⋊1C6, C24⋊3D5, C120⋊3C2, D20⋊1C6, C6.14D20, C30.24D4, C12.53D10, C60.60C22, C5⋊1(C3×D8), C8⋊1(C3×D5), C4.9(C6×D5), (C3×D20)⋊7C2, C20.9(C2×C6), C10.2(C3×D4), C2.4(C3×D20), SmallGroup(240,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D40
G = < a,b,c | a3=b40=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D40
►On 120 points
Generators in S120
(1 74 100)(2 75 101)(3 76 102)(4 77 103)(5 78 104)(6 79 105)(7 80 106)(8 41 107)(9 42 108)(10 43 109)(11 44 110)(12 45 111)(13 46 112)(14 47 113)(15 48 114)(16 49 115)(17 50 116)(18 51 117)(19 52 118)(20 53 119)(21 54 120)(22 55 81)(23 56 82)(24 57 83)(25 58 84)(26 59 85)(27 60 86)(28 61 87)(29 62 88)(30 63 89)(31 64 90)(32 65 91)(33 66 92)(34 67 93)(35 68 94)(36 69 95)(37 70 96)(38 71 97)(39 72 98)(40 73 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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(81 118)(82 117)(83 116)(84 115)(85 114)(86 113)(87 112)(88 111)(89 110)(90 109)(91 108)(92 107)(93 106)(94 105)(95 104)(96 103)(97 102)(98 101)(99 100)(119 120)
G:=sub<Sym(120)| (1,74,100)(2,75,101)(3,76,102)(4,77,103)(5,78,104)(6,79,105)(7,80,106)(8,41,107)(9,42,108)(10,43,109)(11,44,110)(12,45,111)(13,46,112)(14,47,113)(15,48,114)(16,49,115)(17,50,116)(18,51,117)(19,52,118)(20,53,119)(21,54,120)(22,55,81)(23,56,82)(24,57,83)(25,58,84)(26,59,85)(27,60,86)(28,61,87)(29,62,88)(30,63,89)(31,64,90)(32,65,91)(33,66,92)(34,67,93)(35,68,94)(36,69,95)(37,70,96)(38,71,97)(39,72,98)(40,73,99), (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)(119,120)>;
G:=Group( (1,74,100)(2,75,101)(3,76,102)(4,77,103)(5,78,104)(6,79,105)(7,80,106)(8,41,107)(9,42,108)(10,43,109)(11,44,110)(12,45,111)(13,46,112)(14,47,113)(15,48,114)(16,49,115)(17,50,116)(18,51,117)(19,52,118)(20,53,119)(21,54,120)(22,55,81)(23,56,82)(24,57,83)(25,58,84)(26,59,85)(27,60,86)(28,61,87)(29,62,88)(30,63,89)(31,64,90)(32,65,91)(33,66,92)(34,67,93)(35,68,94)(36,69,95)(37,70,96)(38,71,97)(39,72,98)(40,73,99), (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,118)(82,117)(83,116)(84,115)(85,114)(86,113)(87,112)(88,111)(89,110)(90,109)(91,108)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)(119,120) );
G=PermutationGroup([[(1,74,100),(2,75,101),(3,76,102),(4,77,103),(5,78,104),(6,79,105),(7,80,106),(8,41,107),(9,42,108),(10,43,109),(11,44,110),(12,45,111),(13,46,112),(14,47,113),(15,48,114),(16,49,115),(17,50,116),(18,51,117),(19,52,118),(20,53,119),(21,54,120),(22,55,81),(23,56,82),(24,57,83),(25,58,84),(26,59,85),(27,60,86),(28,61,87),(29,62,88),(30,63,89),(31,64,90),(32,65,91),(33,66,92),(34,67,93),(35,68,94),(36,69,95),(37,70,96),(38,71,97),(39,72,98),(40,73,99)], [(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)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(41,66),(42,65),(43,64),(44,63),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(81,118),(82,117),(83,116),(84,115),(85,114),(86,113),(87,112),(88,111),(89,110),(90,109),(91,108),(92,107),(93,106),(94,105),(95,104),(96,103),(97,102),(98,101),(99,100),(119,120)]])
C3×D40 is a maximal subgroup of
C15⋊D16 C3⋊D80 C40.D6 D40.S3 D40⋊S3 C40⋊5D6 C40⋊8D6 D40⋊7S3 D40⋊5S3 C3×D5×D8
69 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 10A | 10B | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 60A | ··· | 60H | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 20 | 20 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
69 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | D5 | D8 | D10 | C3×D4 | C3×D5 | D20 | C3×D8 | C6×D5 | D40 | C3×D20 | C3×D40 |
| kernel | C3×D40 | C120 | C3×D20 | D40 | C40 | D20 | C30 | C24 | C15 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C3×D40 ►in GL2(𝔽241) generated by
| 15 | 0 |
| 0 | 15 |
| 227 | 194 |
| 47 | 20 |
| 227 | 194 |
| 199 | 14 |
G:=sub<GL(2,GF(241))| [15,0,0,15],[227,47,194,20],[227,199,194,14] >;
C3×D40 in GAP, Magma, Sage, TeX
C_3\times D_{40} % in TeX
G:=Group("C3xD40"); // GroupNames label
G:=SmallGroup(240,36);
// by ID
G=gap.SmallGroup(240,36);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-5,169,223,867,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^3=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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