direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D37, C37⋊3C6, C111⋊2C2, SmallGroup(222,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — C3×D37 |
Generators and relations for C3×D37
G = < a,b,c | a3=b37=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D37
►On 111 points
Generators in S111
(1 102 74)(2 103 38)(3 104 39)(4 105 40)(5 106 41)(6 107 42)(7 108 43)(8 109 44)(9 110 45)(10 111 46)(11 75 47)(12 76 48)(13 77 49)(14 78 50)(15 79 51)(16 80 52)(17 81 53)(18 82 54)(19 83 55)(20 84 56)(21 85 57)(22 86 58)(23 87 59)(24 88 60)(25 89 61)(26 90 62)(27 91 63)(28 92 64)(29 93 65)(30 94 66)(31 95 67)(32 96 68)(33 97 69)(34 98 70)(35 99 71)(36 100 72)(37 101 73)
(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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(38 72)(39 71)(40 70)(41 69)(42 68)(43 67)(44 66)(45 65)(46 64)(47 63)(48 62)(49 61)(50 60)(51 59)(52 58)(53 57)(54 56)(73 74)(75 91)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(101 102)
G:=sub<Sym(111)| (1,102,74)(2,103,38)(3,104,39)(4,105,40)(5,106,41)(6,107,42)(7,108,43)(8,109,44)(9,110,45)(10,111,46)(11,75,47)(12,76,48)(13,77,49)(14,78,50)(15,79,51)(16,80,52)(17,81,53)(18,82,54)(19,83,55)(20,84,56)(21,85,57)(22,86,58)(23,87,59)(24,88,60)(25,89,61)(26,90,62)(27,91,63)(28,92,64)(29,93,65)(30,94,66)(31,95,67)(32,96,68)(33,97,69)(34,98,70)(35,99,71)(36,100,72)(37,101,73), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(38,72)(39,71)(40,70)(41,69)(42,68)(43,67)(44,66)(45,65)(46,64)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)(73,74)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)>;
G:=Group( (1,102,74)(2,103,38)(3,104,39)(4,105,40)(5,106,41)(6,107,42)(7,108,43)(8,109,44)(9,110,45)(10,111,46)(11,75,47)(12,76,48)(13,77,49)(14,78,50)(15,79,51)(16,80,52)(17,81,53)(18,82,54)(19,83,55)(20,84,56)(21,85,57)(22,86,58)(23,87,59)(24,88,60)(25,89,61)(26,90,62)(27,91,63)(28,92,64)(29,93,65)(30,94,66)(31,95,67)(32,96,68)(33,97,69)(34,98,70)(35,99,71)(36,100,72)(37,101,73), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)(38,72)(39,71)(40,70)(41,69)(42,68)(43,67)(44,66)(45,65)(46,64)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)(73,74)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102) );
G=PermutationGroup([[(1,102,74),(2,103,38),(3,104,39),(4,105,40),(5,106,41),(6,107,42),(7,108,43),(8,109,44),(9,110,45),(10,111,46),(11,75,47),(12,76,48),(13,77,49),(14,78,50),(15,79,51),(16,80,52),(17,81,53),(18,82,54),(19,83,55),(20,84,56),(21,85,57),(22,86,58),(23,87,59),(24,88,60),(25,89,61),(26,90,62),(27,91,63),(28,92,64),(29,93,65),(30,94,66),(31,95,67),(32,96,68),(33,97,69),(34,98,70),(35,99,71),(36,100,72),(37,101,73)], [(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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,20),(38,72),(39,71),(40,70),(41,69),(42,68),(43,67),(44,66),(45,65),(46,64),(47,63),(48,62),(49,61),(50,60),(51,59),(52,58),(53,57),(54,56),(73,74),(75,91),(76,90),(77,89),(78,88),(79,87),(80,86),(81,85),(82,84),(92,111),(93,110),(94,109),(95,108),(96,107),(97,106),(98,105),(99,104),(100,103),(101,102)]])
C3×D37 is a maximal subgroup of
C37⋊Dic3
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 37A | ··· | 37R | 111A | ··· | 111AJ |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 37 | ··· | 37 | 111 | ··· | 111 |
| size | 1 | 37 | 1 | 1 | 37 | 37 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D37 | C3×D37 |
| kernel | C3×D37 | C111 | D37 | C37 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 18 | 36 |
Matrix representation of C3×D37 ►in GL2(𝔽223) generated by
| 183 | 0 |
| 0 | 183 |
| 44 | 1 |
| 6 | 137 |
| 137 | 222 |
| 36 | 86 |
G:=sub<GL(2,GF(223))| [183,0,0,183],[44,6,1,137],[137,36,222,86] >;
C3×D37 in GAP, Magma, Sage, TeX
C_3\times D_{37} % in TeX
G:=Group("C3xD37"); // GroupNames label
G:=SmallGroup(222,4);
// by ID
G=gap.SmallGroup(222,4);
# by ID
G:=PCGroup([3,-2,-3,-37,1946]);
// Polycyclic
G:=Group<a,b,c|a^3=b^37=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export