metabelian, supersoluble, monomial, A-group
Aliases: C39⋊1Dic3, (C3×C39)⋊2C4, C3⋊(C39⋊C4), C13⋊(C3⋊Dic3), D13.(C3⋊S3), C32⋊3(C13⋊C4), (C3×D13).3S3, (C32×D13).1C2, SmallGroup(468,38)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C39 — C39⋊Dic3 |
Generators and relations for C39⋊Dic3
G = < a,b,c | a39=b6=1, c2=b3, bab-1=a25, cac-1=a8, cbc-1=b-1 >
Smallest permutation representation of C39⋊Dic3
►On 117 points
Generators in S117
(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)
(1 112 73)(2 98 74 26 113 59)(3 84 75 12 114 45)(4 109 76 37 115 70)(5 95 77 23 116 56)(6 81 78 9 117 42)(7 106 40 34 79 67)(8 92 41 20 80 53)(10 103 43 31 82 64)(11 89 44 17 83 50)(13 100 46 28 85 61)(14 86 47)(15 111 48 39 87 72)(16 97 49 25 88 58)(18 108 51 36 90 69)(19 94 52 22 91 55)(21 105 54 33 93 66)(24 102 57 30 96 63)(27 99 60)(29 110 62 38 101 71)(32 107 65 35 104 68)
(2 6 26 9)(3 11 12 17)(4 16 37 25)(5 21 23 33)(7 31 34 10)(8 36 20 18)(13 22 28 19)(14 27)(15 32 39 35)(24 38 30 29)(40 103 67 82)(41 108 53 90)(42 113 78 98)(43 79 64 106)(44 84 50 114)(45 89 75 83)(46 94 61 91)(47 99)(48 104 72 107)(49 109 58 115)(51 80 69 92)(52 85 55 100)(54 95 66 116)(56 105 77 93)(57 110 63 101)(59 81 74 117)(60 86)(62 96 71 102)(65 111 68 87)(70 97 76 88)(73 112)
G:=sub<Sym(117)| (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), (1,112,73)(2,98,74,26,113,59)(3,84,75,12,114,45)(4,109,76,37,115,70)(5,95,77,23,116,56)(6,81,78,9,117,42)(7,106,40,34,79,67)(8,92,41,20,80,53)(10,103,43,31,82,64)(11,89,44,17,83,50)(13,100,46,28,85,61)(14,86,47)(15,111,48,39,87,72)(16,97,49,25,88,58)(18,108,51,36,90,69)(19,94,52,22,91,55)(21,105,54,33,93,66)(24,102,57,30,96,63)(27,99,60)(29,110,62,38,101,71)(32,107,65,35,104,68), (2,6,26,9)(3,11,12,17)(4,16,37,25)(5,21,23,33)(7,31,34,10)(8,36,20,18)(13,22,28,19)(14,27)(15,32,39,35)(24,38,30,29)(40,103,67,82)(41,108,53,90)(42,113,78,98)(43,79,64,106)(44,84,50,114)(45,89,75,83)(46,94,61,91)(47,99)(48,104,72,107)(49,109,58,115)(51,80,69,92)(52,85,55,100)(54,95,66,116)(56,105,77,93)(57,110,63,101)(59,81,74,117)(60,86)(62,96,71,102)(65,111,68,87)(70,97,76,88)(73,112)>;
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), (1,112,73)(2,98,74,26,113,59)(3,84,75,12,114,45)(4,109,76,37,115,70)(5,95,77,23,116,56)(6,81,78,9,117,42)(7,106,40,34,79,67)(8,92,41,20,80,53)(10,103,43,31,82,64)(11,89,44,17,83,50)(13,100,46,28,85,61)(14,86,47)(15,111,48,39,87,72)(16,97,49,25,88,58)(18,108,51,36,90,69)(19,94,52,22,91,55)(21,105,54,33,93,66)(24,102,57,30,96,63)(27,99,60)(29,110,62,38,101,71)(32,107,65,35,104,68), (2,6,26,9)(3,11,12,17)(4,16,37,25)(5,21,23,33)(7,31,34,10)(8,36,20,18)(13,22,28,19)(14,27)(15,32,39,35)(24,38,30,29)(40,103,67,82)(41,108,53,90)(42,113,78,98)(43,79,64,106)(44,84,50,114)(45,89,75,83)(46,94,61,91)(47,99)(48,104,72,107)(49,109,58,115)(51,80,69,92)(52,85,55,100)(54,95,66,116)(56,105,77,93)(57,110,63,101)(59,81,74,117)(60,86)(62,96,71,102)(65,111,68,87)(70,97,76,88)(73,112) );
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)], [(1,112,73),(2,98,74,26,113,59),(3,84,75,12,114,45),(4,109,76,37,115,70),(5,95,77,23,116,56),(6,81,78,9,117,42),(7,106,40,34,79,67),(8,92,41,20,80,53),(10,103,43,31,82,64),(11,89,44,17,83,50),(13,100,46,28,85,61),(14,86,47),(15,111,48,39,87,72),(16,97,49,25,88,58),(18,108,51,36,90,69),(19,94,52,22,91,55),(21,105,54,33,93,66),(24,102,57,30,96,63),(27,99,60),(29,110,62,38,101,71),(32,107,65,35,104,68)], [(2,6,26,9),(3,11,12,17),(4,16,37,25),(5,21,23,33),(7,31,34,10),(8,36,20,18),(13,22,28,19),(14,27),(15,32,39,35),(24,38,30,29),(40,103,67,82),(41,108,53,90),(42,113,78,98),(43,79,64,106),(44,84,50,114),(45,89,75,83),(46,94,61,91),(47,99),(48,104,72,107),(49,109,58,115),(51,80,69,92),(52,85,55,100),(54,95,66,116),(56,105,77,93),(57,110,63,101),(59,81,74,117),(60,86),(62,96,71,102),(65,111,68,87),(70,97,76,88),(73,112)]])
39 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 13A | 13B | 13C | 39A | ··· | 39X |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 13 | 13 | 13 | 39 | ··· | 39 |
| size | 1 | 13 | 2 | 2 | 2 | 2 | 117 | 117 | 26 | 26 | 26 | 26 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | + | ||
| image | C1 | C2 | C4 | S3 | Dic3 | C13⋊C4 | C39⋊C4 |
| kernel | C39⋊Dic3 | C32×D13 | C3×C39 | C3×D13 | C39 | C32 | C3 |
| # reps | 1 | 1 | 2 | 4 | 4 | 3 | 24 |
Matrix representation of C39⋊Dic3 ►in GL6(𝔽157)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 156 | 156 | 0 | 0 | 0 | 0 |
| 0 | 0 | 138 | 155 | 100 | 115 |
| 0 | 0 | 49 | 73 | 23 | 26 |
| 0 | 0 | 6 | 106 | 139 | 47 |
| 0 | 0 | 92 | 140 | 54 | 114 |
| 0 | 156 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 18 | 150 | 87 |
| 0 | 0 | 151 | 64 | 131 | 12 |
| 0 | 0 | 40 | 98 | 145 | 95 |
| 0 | 0 | 120 | 17 | 9 | 104 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 129 | 129 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 146 | 45 | 13 |
| 0 | 0 | 83 | 105 | 101 | 9 |
| 0 | 0 | 40 | 119 | 72 | 84 |
| 0 | 0 | 27 | 24 | 132 | 112 |
G:=sub<GL(6,GF(157))| [0,156,0,0,0,0,1,156,0,0,0,0,0,0,138,49,6,92,0,0,155,73,106,140,0,0,100,23,139,54,0,0,115,26,47,114],[0,1,0,0,0,0,156,1,0,0,0,0,0,0,1,151,40,120,0,0,18,64,98,17,0,0,150,131,145,9,0,0,87,12,95,104],[28,129,0,0,0,0,0,129,0,0,0,0,0,0,25,83,40,27,0,0,146,105,119,24,0,0,45,101,72,132,0,0,13,9,84,112] >;
C39⋊Dic3 in GAP, Magma, Sage, TeX
C_{39}\rtimes {\rm Dic}_3 % in TeX
G:=Group("C39:Dic3"); // GroupNames label
G:=SmallGroup(468,38);
// by ID
G=gap.SmallGroup(468,38);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-13,10,122,483,4504,5409]);
// Polycyclic
G:=Group<a,b,c|a^39=b^6=1,c^2=b^3,b*a*b^-1=a^25,c*a*c^-1=a^8,c*b*c^-1=b^-1>;
// generators/relations
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