metabelian, supersoluble, monomial, A-group
Aliases: C3⋊D39, C39⋊1S3, C32⋊2D13, C13⋊(C3⋊S3), (C3×C39)⋊1C2, SmallGroup(234,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C39 — C3⋊D39 |
Generators and relations for C3⋊D39
G = < a,b,c | a3=b39=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C3⋊D39
►On 117 points
Generators in S117
(1 93 68)(2 94 69)(3 95 70)(4 96 71)(5 97 72)(6 98 73)(7 99 74)(8 100 75)(9 101 76)(10 102 77)(11 103 78)(12 104 40)(13 105 41)(14 106 42)(15 107 43)(16 108 44)(17 109 45)(18 110 46)(19 111 47)(20 112 48)(21 113 49)(22 114 50)(23 115 51)(24 116 52)(25 117 53)(26 79 54)(27 80 55)(28 81 56)(29 82 57)(30 83 58)(31 84 59)(32 85 60)(33 86 61)(34 87 62)(35 88 63)(36 89 64)(37 90 65)(38 91 66)(39 92 67)
(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)
(1 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(40 81)(41 80)(42 79)(43 117)(44 116)(45 115)(46 114)(47 113)(48 112)(49 111)(50 110)(51 109)(52 108)(53 107)(54 106)(55 105)(56 104)(57 103)(58 102)(59 101)(60 100)(61 99)(62 98)(63 97)(64 96)(65 95)(66 94)(67 93)(68 92)(69 91)(70 90)(71 89)(72 88)(73 87)(74 86)(75 85)(76 84)(77 83)(78 82)
G:=sub<Sym(117)| (1,93,68)(2,94,69)(3,95,70)(4,96,71)(5,97,72)(6,98,73)(7,99,74)(8,100,75)(9,101,76)(10,102,77)(11,103,78)(12,104,40)(13,105,41)(14,106,42)(15,107,43)(16,108,44)(17,109,45)(18,110,46)(19,111,47)(20,112,48)(21,113,49)(22,114,50)(23,115,51)(24,116,52)(25,117,53)(26,79,54)(27,80,55)(28,81,56)(29,82,57)(30,83,58)(31,84,59)(32,85,60)(33,86,61)(34,87,62)(35,88,63)(36,89,64)(37,90,65)(38,91,66)(39,92,67), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(40,81)(41,80)(42,79)(43,117)(44,116)(45,115)(46,114)(47,113)(48,112)(49,111)(50,110)(51,109)(52,108)(53,107)(54,106)(55,105)(56,104)(57,103)(58,102)(59,101)(60,100)(61,99)(62,98)(63,97)(64,96)(65,95)(66,94)(67,93)(68,92)(69,91)(70,90)(71,89)(72,88)(73,87)(74,86)(75,85)(76,84)(77,83)(78,82)>;
G:=Group( (1,93,68)(2,94,69)(3,95,70)(4,96,71)(5,97,72)(6,98,73)(7,99,74)(8,100,75)(9,101,76)(10,102,77)(11,103,78)(12,104,40)(13,105,41)(14,106,42)(15,107,43)(16,108,44)(17,109,45)(18,110,46)(19,111,47)(20,112,48)(21,113,49)(22,114,50)(23,115,51)(24,116,52)(25,117,53)(26,79,54)(27,80,55)(28,81,56)(29,82,57)(30,83,58)(31,84,59)(32,85,60)(33,86,61)(34,87,62)(35,88,63)(36,89,64)(37,90,65)(38,91,66)(39,92,67), (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(40,81)(41,80)(42,79)(43,117)(44,116)(45,115)(46,114)(47,113)(48,112)(49,111)(50,110)(51,109)(52,108)(53,107)(54,106)(55,105)(56,104)(57,103)(58,102)(59,101)(60,100)(61,99)(62,98)(63,97)(64,96)(65,95)(66,94)(67,93)(68,92)(69,91)(70,90)(71,89)(72,88)(73,87)(74,86)(75,85)(76,84)(77,83)(78,82) );
G=PermutationGroup([[(1,93,68),(2,94,69),(3,95,70),(4,96,71),(5,97,72),(6,98,73),(7,99,74),(8,100,75),(9,101,76),(10,102,77),(11,103,78),(12,104,40),(13,105,41),(14,106,42),(15,107,43),(16,108,44),(17,109,45),(18,110,46),(19,111,47),(20,112,48),(21,113,49),(22,114,50),(23,115,51),(24,116,52),(25,117,53),(26,79,54),(27,80,55),(28,81,56),(29,82,57),(30,83,58),(31,84,59),(32,85,60),(33,86,61),(34,87,62),(35,88,63),(36,89,64),(37,90,65),(38,91,66),(39,92,67)], [(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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(16,24),(17,23),(18,22),(19,21),(40,81),(41,80),(42,79),(43,117),(44,116),(45,115),(46,114),(47,113),(48,112),(49,111),(50,110),(51,109),(52,108),(53,107),(54,106),(55,105),(56,104),(57,103),(58,102),(59,101),(60,100),(61,99),(62,98),(63,97),(64,96),(65,95),(66,94),(67,93),(68,92),(69,91),(70,90),(71,89),(72,88),(73,87),(74,86),(75,85),(76,84),(77,83),(78,82)]])
C3⋊D39 is a maximal subgroup of
(C3×C39)⋊C4 C3⋊S3×D13 S3×D39
C3⋊D39 is a maximal quotient of C3⋊Dic39
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 13A | ··· | 13F | 39A | ··· | 39AV |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 13 | ··· | 13 | 39 | ··· | 39 |
| size | 1 | 117 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | S3 | D13 | D39 |
| kernel | C3⋊D39 | C3×C39 | C39 | C32 | C3 |
| # reps | 1 | 1 | 4 | 6 | 48 |
Matrix representation of C3⋊D39 ►in GL4(𝔽79) generated by
| 49 | 71 | 0 | 0 |
| 20 | 29 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 34 | 70 | 0 | 0 |
| 62 | 51 | 0 | 0 |
| 0 | 0 | 19 | 19 |
| 0 | 0 | 77 | 23 |
| 11 | 25 | 0 | 0 |
| 11 | 68 | 0 | 0 |
| 0 | 0 | 65 | 3 |
| 0 | 0 | 14 | 14 |
G:=sub<GL(4,GF(79))| [49,20,0,0,71,29,0,0,0,0,1,0,0,0,0,1],[34,62,0,0,70,51,0,0,0,0,19,77,0,0,19,23],[11,11,0,0,25,68,0,0,0,0,65,14,0,0,3,14] >;
C3⋊D39 in GAP, Magma, Sage, TeX
C_3\rtimes D_{39} % in TeX
G:=Group("C3:D39"); // GroupNames label
G:=SmallGroup(234,15);
// by ID
G=gap.SmallGroup(234,15);
# by ID
G:=PCGroup([4,-2,-3,-3,-13,33,146,3459]);
// Polycyclic
G:=Group<a,b,c|a^3=b^39=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export