metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊2D8, C3⋊2D40, D60⋊5C2, D20⋊1S3, C6.6D20, C30.2D4, C20.22D6, C12.2D10, C60.8C22, C3⋊C8⋊1D5, C5⋊1(D4⋊S3), C4.1(S3×D5), (C3×D20)⋊1C2, C10.1(C3⋊D4), C2.4(C3⋊D20), (C5×C3⋊C8)⋊1C2, SmallGroup(240,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊D40
G = < a,b,c | a3=b40=c2=1, bab-1=cac=a-1, cbc=b-1 >
20C2
60C2
10C22
30C22
20C6
20S3
4D5
12D5
3C8
5D4
15D4
10C2×C6
10D6
2D10
6D10
4D15
15D8
5D12
3C40
3D20
2D30
3D40
Smallest permutation representation of C3⋊D40
►On 120 points
Generators in S120
(1 116 80)(2 41 117)(3 118 42)(4 43 119)(5 120 44)(6 45 81)(7 82 46)(8 47 83)(9 84 48)(10 49 85)(11 86 50)(12 51 87)(13 88 52)(14 53 89)(15 90 54)(16 55 91)(17 92 56)(18 57 93)(19 94 58)(20 59 95)(21 96 60)(22 61 97)(23 98 62)(24 63 99)(25 100 64)(26 65 101)(27 102 66)(28 67 103)(29 104 68)(30 69 105)(31 106 70)(32 71 107)(33 108 72)(34 73 109)(35 110 74)(36 75 111)(37 112 76)(38 77 113)(39 114 78)(40 79 115)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 89)(42 88)(43 87)(44 86)(45 85)(46 84)(47 83)(48 82)(49 81)(50 120)(51 119)(52 118)(53 117)(54 116)(55 115)(56 114)(57 113)(58 112)(59 111)(60 110)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)
G:=sub<Sym(120)| (1,116,80)(2,41,117)(3,118,42)(4,43,119)(5,120,44)(6,45,81)(7,82,46)(8,47,83)(9,84,48)(10,49,85)(11,86,50)(12,51,87)(13,88,52)(14,53,89)(15,90,54)(16,55,91)(17,92,56)(18,57,93)(19,94,58)(20,59,95)(21,96,60)(22,61,97)(23,98,62)(24,63,99)(25,100,64)(26,65,101)(27,102,66)(28,67,103)(29,104,68)(30,69,105)(31,106,70)(32,71,107)(33,108,72)(34,73,109)(35,110,74)(36,75,111)(37,112,76)(38,77,113)(39,114,78)(40,79,115), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,81)(50,120)(51,119)(52,118)(53,117)(54,116)(55,115)(56,114)(57,113)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)>;
G:=Group( (1,116,80)(2,41,117)(3,118,42)(4,43,119)(5,120,44)(6,45,81)(7,82,46)(8,47,83)(9,84,48)(10,49,85)(11,86,50)(12,51,87)(13,88,52)(14,53,89)(15,90,54)(16,55,91)(17,92,56)(18,57,93)(19,94,58)(20,59,95)(21,96,60)(22,61,97)(23,98,62)(24,63,99)(25,100,64)(26,65,101)(27,102,66)(28,67,103)(29,104,68)(30,69,105)(31,106,70)(32,71,107)(33,108,72)(34,73,109)(35,110,74)(36,75,111)(37,112,76)(38,77,113)(39,114,78)(40,79,115), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,81)(50,120)(51,119)(52,118)(53,117)(54,116)(55,115)(56,114)(57,113)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90) );
G=PermutationGroup([[(1,116,80),(2,41,117),(3,118,42),(4,43,119),(5,120,44),(6,45,81),(7,82,46),(8,47,83),(9,84,48),(10,49,85),(11,86,50),(12,51,87),(13,88,52),(14,53,89),(15,90,54),(16,55,91),(17,92,56),(18,57,93),(19,94,58),(20,59,95),(21,96,60),(22,61,97),(23,98,62),(24,63,99),(25,100,64),(26,65,101),(27,102,66),(28,67,103),(29,104,68),(30,69,105),(31,106,70),(32,71,107),(33,108,72),(34,73,109),(35,110,74),(36,75,111),(37,112,76),(38,77,113),(39,114,78),(40,79,115)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(41,89),(42,88),(43,87),(44,86),(45,85),(46,84),(47,83),(48,82),(49,81),(50,120),(51,119),(52,118),(53,117),(54,116),(55,115),(56,114),(57,113),(58,112),(59,111),(60,110),(61,109),(62,108),(63,107),(64,106),(65,105),(66,104),(67,103),(68,102),(69,101),(70,100),(71,99),(72,98),(73,97),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90)]])
C3⋊D40 is a maximal subgroup of
S3×D40 C40⋊1D6 D40⋊S3 D6.1D20 D20⋊19D6 D20.31D6 D60⋊30C22 D5×D4⋊S3 D15⋊D8 D20.9D6 Dic6⋊D10 D20⋊D6 D60⋊C22 D20.D6 D20.16D6
C3⋊D40 is a maximal quotient of
C3⋊D80 D40.S3 C24.D10 C3⋊Dic40 C6.D40 D60⋊12C4 C60.5Q8
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 12 | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | 40A | ··· | 40H | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 40 | ··· | 40 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 20 | 60 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 6 | 6 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D8 | D10 | C3⋊D4 | D20 | D40 | D4⋊S3 | S3×D5 | C3⋊D20 | C3⋊D40 |
| kernel | C3⋊D40 | C5×C3⋊C8 | C3×D20 | D60 | D20 | C30 | C3⋊C8 | C20 | C15 | C12 | C10 | C6 | C3 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of C3⋊D40 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 1 |
| 0 | 0 | 240 | 0 |
| 56 | 27 | 0 | 0 |
| 214 | 228 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 78 | 163 | 0 | 0 |
| 44 | 163 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,240,240,0,0,1,0],[56,214,0,0,27,228,0,0,0,0,0,1,0,0,1,0],[78,44,0,0,163,163,0,0,0,0,0,1,0,0,1,0] >;
C3⋊D40 in GAP, Magma, Sage, TeX
C_3\rtimes D_{40} % in TeX
G:=Group("C3:D40"); // GroupNames label
G:=SmallGroup(240,14);
// by ID
G=gap.SmallGroup(240,14);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,73,79,218,50,490,6917]);
// Polycyclic
G:=Group<a,b,c|a^3=b^40=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export