direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D31, C31⋊3C6, C93⋊2C2, SmallGroup(186,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C3×D31 |
Generators and relations for C3×D31
G = < a,b,c | a3=b31=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D31
►On 93 points
Generators in S93
(1 86 32)(2 87 33)(3 88 34)(4 89 35)(5 90 36)(6 91 37)(7 92 38)(8 93 39)(9 63 40)(10 64 41)(11 65 42)(12 66 43)(13 67 44)(14 68 45)(15 69 46)(16 70 47)(17 71 48)(18 72 49)(19 73 50)(20 74 51)(21 75 52)(22 76 53)(23 77 54)(24 78 55)(25 79 56)(26 80 57)(27 81 58)(28 82 59)(29 83 60)(30 84 61)(31 85 62)
(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)
(1 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(32 62)(33 61)(34 60)(35 59)(36 58)(37 57)(38 56)(39 55)(40 54)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(78 93)(79 92)(80 91)(81 90)(82 89)(83 88)(84 87)(85 86)
G:=sub<Sym(93)| (1,86,32)(2,87,33)(3,88,34)(4,89,35)(5,90,36)(6,91,37)(7,92,38)(8,93,39)(9,63,40)(10,64,41)(11,65,42)(12,66,43)(13,67,44)(14,68,45)(15,69,46)(16,70,47)(17,71,48)(18,72,49)(19,73,50)(20,74,51)(21,75,52)(22,76,53)(23,77,54)(24,78,55)(25,79,56)(26,80,57)(27,81,58)(28,82,59)(29,83,60)(30,84,61)(31,85,62), (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(32,62)(33,61)(34,60)(35,59)(36,58)(37,57)(38,56)(39,55)(40,54)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)>;
G:=Group( (1,86,32)(2,87,33)(3,88,34)(4,89,35)(5,90,36)(6,91,37)(7,92,38)(8,93,39)(9,63,40)(10,64,41)(11,65,42)(12,66,43)(13,67,44)(14,68,45)(15,69,46)(16,70,47)(17,71,48)(18,72,49)(19,73,50)(20,74,51)(21,75,52)(22,76,53)(23,77,54)(24,78,55)(25,79,56)(26,80,57)(27,81,58)(28,82,59)(29,83,60)(30,84,61)(31,85,62), (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(32,62)(33,61)(34,60)(35,59)(36,58)(37,57)(38,56)(39,55)(40,54)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86) );
G=PermutationGroup([[(1,86,32),(2,87,33),(3,88,34),(4,89,35),(5,90,36),(6,91,37),(7,92,38),(8,93,39),(9,63,40),(10,64,41),(11,65,42),(12,66,43),(13,67,44),(14,68,45),(15,69,46),(16,70,47),(17,71,48),(18,72,49),(19,73,50),(20,74,51),(21,75,52),(22,76,53),(23,77,54),(24,78,55),(25,79,56),(26,80,57),(27,81,58),(28,82,59),(29,83,60),(30,84,61),(31,85,62)], [(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)], [(1,31),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,21),(12,20),(13,19),(14,18),(15,17),(32,62),(33,61),(34,60),(35,59),(36,58),(37,57),(38,56),(39,55),(40,54),(41,53),(42,52),(43,51),(44,50),(45,49),(46,48),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(78,93),(79,92),(80,91),(81,90),(82,89),(83,88),(84,87),(85,86)]])
51 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 31A | ··· | 31O | 93A | ··· | 93AD |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 31 | ··· | 31 | 93 | ··· | 93 |
| size | 1 | 31 | 1 | 1 | 31 | 31 | 2 | ··· | 2 | 2 | ··· | 2 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D31 | C3×D31 |
| kernel | C3×D31 | C93 | D31 | C31 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 15 | 30 |
Matrix representation of C3×D31 ►in GL2(𝔽373) generated by
| 88 | 0 |
| 0 | 88 |
| 178 | 1 |
| 318 | 138 |
| 34 | 316 |
| 354 | 339 |
G:=sub<GL(2,GF(373))| [88,0,0,88],[178,318,1,138],[34,354,316,339] >;
C3×D31 in GAP, Magma, Sage, TeX
C_3\times D_{31} % in TeX
G:=Group("C3xD31"); // GroupNames label
G:=SmallGroup(186,4);
// by ID
G=gap.SmallGroup(186,4);
# by ID
G:=PCGroup([3,-2,-3,-31,1622]);
// Polycyclic
G:=Group<a,b,c|a^3=b^31=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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