direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary
Aliases: C3×C37⋊C3, C111⋊C3, C37⋊C32, SmallGroup(333,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — C3×C37⋊C3 |
Generators and relations for C3×C37⋊C3
G = < a,b,c | a3=b37=c3=1, ab=ba, ac=ca, cbc-1=b10 >
Smallest permutation representation of C3×C37⋊C3
►On 111 points
Generators in S111
(1 75 38)(2 76 39)(3 77 40)(4 78 41)(5 79 42)(6 80 43)(7 81 44)(8 82 45)(9 83 46)(10 84 47)(11 85 48)(12 86 49)(13 87 50)(14 88 51)(15 89 52)(16 90 53)(17 91 54)(18 92 55)(19 93 56)(20 94 57)(21 95 58)(22 96 59)(23 97 60)(24 98 61)(25 99 62)(26 100 63)(27 101 64)(28 102 65)(29 103 66)(30 104 67)(31 105 68)(32 106 69)(33 107 70)(34 108 71)(35 109 72)(36 110 73)(37 111 74)
(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)
(2 27 11)(3 16 21)(4 5 31)(6 20 14)(7 9 24)(8 35 34)(10 13 17)(12 28 37)(15 32 30)(18 36 23)(19 25 33)(22 29 26)(39 64 48)(40 53 58)(41 42 68)(43 57 51)(44 46 61)(45 72 71)(47 50 54)(49 65 74)(52 69 67)(55 73 60)(56 62 70)(59 66 63)(76 101 85)(77 90 95)(78 79 105)(80 94 88)(81 83 98)(82 109 108)(84 87 91)(86 102 111)(89 106 104)(92 110 97)(93 99 107)(96 103 100)
G:=sub<Sym(111)| (1,75,38)(2,76,39)(3,77,40)(4,78,41)(5,79,42)(6,80,43)(7,81,44)(8,82,45)(9,83,46)(10,84,47)(11,85,48)(12,86,49)(13,87,50)(14,88,51)(15,89,52)(16,90,53)(17,91,54)(18,92,55)(19,93,56)(20,94,57)(21,95,58)(22,96,59)(23,97,60)(24,98,61)(25,99,62)(26,100,63)(27,101,64)(28,102,65)(29,103,66)(30,104,67)(31,105,68)(32,106,69)(33,107,70)(34,108,71)(35,109,72)(36,110,73)(37,111,74), (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), (2,27,11)(3,16,21)(4,5,31)(6,20,14)(7,9,24)(8,35,34)(10,13,17)(12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)(39,64,48)(40,53,58)(41,42,68)(43,57,51)(44,46,61)(45,72,71)(47,50,54)(49,65,74)(52,69,67)(55,73,60)(56,62,70)(59,66,63)(76,101,85)(77,90,95)(78,79,105)(80,94,88)(81,83,98)(82,109,108)(84,87,91)(86,102,111)(89,106,104)(92,110,97)(93,99,107)(96,103,100)>;
G:=Group( (1,75,38)(2,76,39)(3,77,40)(4,78,41)(5,79,42)(6,80,43)(7,81,44)(8,82,45)(9,83,46)(10,84,47)(11,85,48)(12,86,49)(13,87,50)(14,88,51)(15,89,52)(16,90,53)(17,91,54)(18,92,55)(19,93,56)(20,94,57)(21,95,58)(22,96,59)(23,97,60)(24,98,61)(25,99,62)(26,100,63)(27,101,64)(28,102,65)(29,103,66)(30,104,67)(31,105,68)(32,106,69)(33,107,70)(34,108,71)(35,109,72)(36,110,73)(37,111,74), (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), (2,27,11)(3,16,21)(4,5,31)(6,20,14)(7,9,24)(8,35,34)(10,13,17)(12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)(39,64,48)(40,53,58)(41,42,68)(43,57,51)(44,46,61)(45,72,71)(47,50,54)(49,65,74)(52,69,67)(55,73,60)(56,62,70)(59,66,63)(76,101,85)(77,90,95)(78,79,105)(80,94,88)(81,83,98)(82,109,108)(84,87,91)(86,102,111)(89,106,104)(92,110,97)(93,99,107)(96,103,100) );
G=PermutationGroup([[(1,75,38),(2,76,39),(3,77,40),(4,78,41),(5,79,42),(6,80,43),(7,81,44),(8,82,45),(9,83,46),(10,84,47),(11,85,48),(12,86,49),(13,87,50),(14,88,51),(15,89,52),(16,90,53),(17,91,54),(18,92,55),(19,93,56),(20,94,57),(21,95,58),(22,96,59),(23,97,60),(24,98,61),(25,99,62),(26,100,63),(27,101,64),(28,102,65),(29,103,66),(30,104,67),(31,105,68),(32,106,69),(33,107,70),(34,108,71),(35,109,72),(36,110,73),(37,111,74)], [(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)], [(2,27,11),(3,16,21),(4,5,31),(6,20,14),(7,9,24),(8,35,34),(10,13,17),(12,28,37),(15,32,30),(18,36,23),(19,25,33),(22,29,26),(39,64,48),(40,53,58),(41,42,68),(43,57,51),(44,46,61),(45,72,71),(47,50,54),(49,65,74),(52,69,67),(55,73,60),(56,62,70),(59,66,63),(76,101,85),(77,90,95),(78,79,105),(80,94,88),(81,83,98),(82,109,108),(84,87,91),(86,102,111),(89,106,104),(92,110,97),(93,99,107),(96,103,100)]])
45 conjugacy classes
| class | 1 | 3A | 3B | 3C | ··· | 3H | 37A | ··· | 37L | 111A | ··· | 111X |
| order | 1 | 3 | 3 | 3 | ··· | 3 | 37 | ··· | 37 | 111 | ··· | 111 |
| size | 1 | 1 | 1 | 37 | ··· | 37 | 3 | ··· | 3 | 3 | ··· | 3 |
45 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 |
| type | + | ||||
| image | C1 | C3 | C3 | C37⋊C3 | C3×C37⋊C3 |
| kernel | C3×C37⋊C3 | C37⋊C3 | C111 | C3 | C1 |
| # reps | 1 | 6 | 2 | 12 | 24 |
Matrix representation of C3×C37⋊C3 ►in GL3(𝔽223) generated by
| 39 | 0 | 0 |
| 0 | 39 | 0 |
| 0 | 0 | 39 |
| 42 | 147 | 217 |
| 1 | 0 | 8 |
| 0 | 1 | 72 |
| 49 | 32 | 20 |
| 182 | 23 | 73 |
| 64 | 64 | 151 |
G:=sub<GL(3,GF(223))| [39,0,0,0,39,0,0,0,39],[42,1,0,147,0,1,217,8,72],[49,182,64,32,23,64,20,73,151] >;
C3×C37⋊C3 in GAP, Magma, Sage, TeX
C_3\times C_{37}\rtimes C_3 % in TeX
G:=Group("C3xC37:C3"); // GroupNames label
G:=SmallGroup(333,4);
// by ID
G=gap.SmallGroup(333,4);
# by ID
G:=PCGroup([3,-3,-3,-37,2108]);
// Polycyclic
G:=Group<a,b,c|a^3=b^37=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations
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