metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C37⋊Dic3, C111⋊1C4, D37.S3, C3⋊(C37⋊C4), (C3×D37).1C2, SmallGroup(444,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C111 — C37⋊Dic3 |
Generators and relations for C37⋊Dic3
G = < a,b,c | a37=b6=1, c2=b3, bab-1=a-1, cac-1=a31, cbc-1=b-1 >
Smallest permutation representation of C37⋊Dic3
►On 111 points
Generators in S111
(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)
(1 83 39)(2 82 40 37 84 38)(3 81 41 36 85 74)(4 80 42 35 86 73)(5 79 43 34 87 72)(6 78 44 33 88 71)(7 77 45 32 89 70)(8 76 46 31 90 69)(9 75 47 30 91 68)(10 111 48 29 92 67)(11 110 49 28 93 66)(12 109 50 27 94 65)(13 108 51 26 95 64)(14 107 52 25 96 63)(15 106 53 24 97 62)(16 105 54 23 98 61)(17 104 55 22 99 60)(18 103 56 21 100 59)(19 102 57 20 101 58)
(2 7 37 32)(3 13 36 26)(4 19 35 20)(5 25 34 14)(6 31 33 8)(9 12 30 27)(10 18 29 21)(11 24 28 15)(16 17 23 22)(38 77 40 89)(39 83)(41 95 74 108)(42 101 73 102)(43 107 72 96)(44 76 71 90)(45 82 70 84)(46 88 69 78)(47 94 68 109)(48 100 67 103)(49 106 66 97)(50 75 65 91)(51 81 64 85)(52 87 63 79)(53 93 62 110)(54 99 61 104)(55 105 60 98)(56 111 59 92)(57 80 58 86)
G:=sub<Sym(111)| (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), (1,83,39)(2,82,40,37,84,38)(3,81,41,36,85,74)(4,80,42,35,86,73)(5,79,43,34,87,72)(6,78,44,33,88,71)(7,77,45,32,89,70)(8,76,46,31,90,69)(9,75,47,30,91,68)(10,111,48,29,92,67)(11,110,49,28,93,66)(12,109,50,27,94,65)(13,108,51,26,95,64)(14,107,52,25,96,63)(15,106,53,24,97,62)(16,105,54,23,98,61)(17,104,55,22,99,60)(18,103,56,21,100,59)(19,102,57,20,101,58), (2,7,37,32)(3,13,36,26)(4,19,35,20)(5,25,34,14)(6,31,33,8)(9,12,30,27)(10,18,29,21)(11,24,28,15)(16,17,23,22)(38,77,40,89)(39,83)(41,95,74,108)(42,101,73,102)(43,107,72,96)(44,76,71,90)(45,82,70,84)(46,88,69,78)(47,94,68,109)(48,100,67,103)(49,106,66,97)(50,75,65,91)(51,81,64,85)(52,87,63,79)(53,93,62,110)(54,99,61,104)(55,105,60,98)(56,111,59,92)(57,80,58,86)>;
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), (1,83,39)(2,82,40,37,84,38)(3,81,41,36,85,74)(4,80,42,35,86,73)(5,79,43,34,87,72)(6,78,44,33,88,71)(7,77,45,32,89,70)(8,76,46,31,90,69)(9,75,47,30,91,68)(10,111,48,29,92,67)(11,110,49,28,93,66)(12,109,50,27,94,65)(13,108,51,26,95,64)(14,107,52,25,96,63)(15,106,53,24,97,62)(16,105,54,23,98,61)(17,104,55,22,99,60)(18,103,56,21,100,59)(19,102,57,20,101,58), (2,7,37,32)(3,13,36,26)(4,19,35,20)(5,25,34,14)(6,31,33,8)(9,12,30,27)(10,18,29,21)(11,24,28,15)(16,17,23,22)(38,77,40,89)(39,83)(41,95,74,108)(42,101,73,102)(43,107,72,96)(44,76,71,90)(45,82,70,84)(46,88,69,78)(47,94,68,109)(48,100,67,103)(49,106,66,97)(50,75,65,91)(51,81,64,85)(52,87,63,79)(53,93,62,110)(54,99,61,104)(55,105,60,98)(56,111,59,92)(57,80,58,86) );
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)], [(1,83,39),(2,82,40,37,84,38),(3,81,41,36,85,74),(4,80,42,35,86,73),(5,79,43,34,87,72),(6,78,44,33,88,71),(7,77,45,32,89,70),(8,76,46,31,90,69),(9,75,47,30,91,68),(10,111,48,29,92,67),(11,110,49,28,93,66),(12,109,50,27,94,65),(13,108,51,26,95,64),(14,107,52,25,96,63),(15,106,53,24,97,62),(16,105,54,23,98,61),(17,104,55,22,99,60),(18,103,56,21,100,59),(19,102,57,20,101,58)], [(2,7,37,32),(3,13,36,26),(4,19,35,20),(5,25,34,14),(6,31,33,8),(9,12,30,27),(10,18,29,21),(11,24,28,15),(16,17,23,22),(38,77,40,89),(39,83),(41,95,74,108),(42,101,73,102),(43,107,72,96),(44,76,71,90),(45,82,70,84),(46,88,69,78),(47,94,68,109),(48,100,67,103),(49,106,66,97),(50,75,65,91),(51,81,64,85),(52,87,63,79),(53,93,62,110),(54,99,61,104),(55,105,60,98),(56,111,59,92),(57,80,58,86)]])
33 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 37A | ··· | 37I | 111A | ··· | 111R |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 37 | ··· | 37 | 111 | ··· | 111 |
| size | 1 | 37 | 2 | 111 | 111 | 74 | 4 | ··· | 4 | 4 | ··· | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | + | ||
| image | C1 | C2 | C4 | S3 | Dic3 | C37⋊C4 | C37⋊Dic3 |
| kernel | C37⋊Dic3 | C3×D37 | C111 | D37 | C37 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 9 | 18 |
Matrix representation of C37⋊Dic3 ►in GL4(𝔽1777) generated by
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1776 | 1020 | 1014 | 1020 |
| 527 | 623 | 1055 | 269 |
| 1509 | 556 | 1155 | 1250 |
| 854 | 1284 | 1548 | 268 |
| 1634 | 328 | 618 | 923 |
| 1 | 0 | 0 | 0 |
| 92 | 346 | 796 | 1033 |
| 686 | 1180 | 1538 | 1073 |
| 1578 | 930 | 94 | 1669 |
G:=sub<GL(4,GF(1777))| [0,0,0,1776,1,0,0,1020,0,1,0,1014,0,0,1,1020],[527,1509,854,1634,623,556,1284,328,1055,1155,1548,618,269,1250,268,923],[1,92,686,1578,0,346,1180,930,0,796,1538,94,0,1033,1073,1669] >;
C37⋊Dic3 in GAP, Magma, Sage, TeX
C_{37}\rtimes {\rm Dic}_3 % in TeX
G:=Group("C37:Dic3"); // GroupNames label
G:=SmallGroup(444,10);
// by ID
G=gap.SmallGroup(444,10);
# by ID
G:=PCGroup([4,-2,-2,-3,-37,8,98,1155,3463]);
// Polycyclic
G:=Group<a,b,c|a^37=b^6=1,c^2=b^3,b*a*b^-1=a^-1,c*a*c^-1=a^31,c*b*c^-1=b^-1>;
// generators/relations
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