direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary
Aliases: C3×C31⋊C3, C93⋊C3, C31⋊C32, SmallGroup(279,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C3×C31⋊C3 |
Generators and relations for C3×C31⋊C3
G = < a,b,c | a3=b31=c3=1, ab=ba, ac=ca, cbc-1=b5 >
Smallest permutation representation of C3×C31⋊C3
►On 93 points
Generators in S93
(1 63 32)(2 64 33)(3 65 34)(4 66 35)(5 67 36)(6 68 37)(7 69 38)(8 70 39)(9 71 40)(10 72 41)(11 73 42)(12 74 43)(13 75 44)(14 76 45)(15 77 46)(16 78 47)(17 79 48)(18 80 49)(19 81 50)(20 82 51)(21 83 52)(22 84 53)(23 85 54)(24 86 55)(25 87 56)(26 88 57)(27 89 58)(28 90 59)(29 91 60)(30 92 61)(31 93 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)
(2 26 6)(3 20 11)(4 14 16)(5 8 21)(7 27 31)(9 15 10)(12 28 25)(13 22 30)(17 29 19)(18 23 24)(33 57 37)(34 51 42)(35 45 47)(36 39 52)(38 58 62)(40 46 41)(43 59 56)(44 53 61)(48 60 50)(49 54 55)(64 88 68)(65 82 73)(66 76 78)(67 70 83)(69 89 93)(71 77 72)(74 90 87)(75 84 92)(79 91 81)(80 85 86)
G:=sub<Sym(93)| (1,63,32)(2,64,33)(3,65,34)(4,66,35)(5,67,36)(6,68,37)(7,69,38)(8,70,39)(9,71,40)(10,72,41)(11,73,42)(12,74,43)(13,75,44)(14,76,45)(15,77,46)(16,78,47)(17,79,48)(18,80,49)(19,81,50)(20,82,51)(21,83,52)(22,84,53)(23,85,54)(24,86,55)(25,87,56)(26,88,57)(27,89,58)(28,90,59)(29,91,60)(30,92,61)(31,93,62), (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), (2,26,6)(3,20,11)(4,14,16)(5,8,21)(7,27,31)(9,15,10)(12,28,25)(13,22,30)(17,29,19)(18,23,24)(33,57,37)(34,51,42)(35,45,47)(36,39,52)(38,58,62)(40,46,41)(43,59,56)(44,53,61)(48,60,50)(49,54,55)(64,88,68)(65,82,73)(66,76,78)(67,70,83)(69,89,93)(71,77,72)(74,90,87)(75,84,92)(79,91,81)(80,85,86)>;
G:=Group( (1,63,32)(2,64,33)(3,65,34)(4,66,35)(5,67,36)(6,68,37)(7,69,38)(8,70,39)(9,71,40)(10,72,41)(11,73,42)(12,74,43)(13,75,44)(14,76,45)(15,77,46)(16,78,47)(17,79,48)(18,80,49)(19,81,50)(20,82,51)(21,83,52)(22,84,53)(23,85,54)(24,86,55)(25,87,56)(26,88,57)(27,89,58)(28,90,59)(29,91,60)(30,92,61)(31,93,62), (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), (2,26,6)(3,20,11)(4,14,16)(5,8,21)(7,27,31)(9,15,10)(12,28,25)(13,22,30)(17,29,19)(18,23,24)(33,57,37)(34,51,42)(35,45,47)(36,39,52)(38,58,62)(40,46,41)(43,59,56)(44,53,61)(48,60,50)(49,54,55)(64,88,68)(65,82,73)(66,76,78)(67,70,83)(69,89,93)(71,77,72)(74,90,87)(75,84,92)(79,91,81)(80,85,86) );
G=PermutationGroup([[(1,63,32),(2,64,33),(3,65,34),(4,66,35),(5,67,36),(6,68,37),(7,69,38),(8,70,39),(9,71,40),(10,72,41),(11,73,42),(12,74,43),(13,75,44),(14,76,45),(15,77,46),(16,78,47),(17,79,48),(18,80,49),(19,81,50),(20,82,51),(21,83,52),(22,84,53),(23,85,54),(24,86,55),(25,87,56),(26,88,57),(27,89,58),(28,90,59),(29,91,60),(30,92,61),(31,93,62)], [(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)], [(2,26,6),(3,20,11),(4,14,16),(5,8,21),(7,27,31),(9,15,10),(12,28,25),(13,22,30),(17,29,19),(18,23,24),(33,57,37),(34,51,42),(35,45,47),(36,39,52),(38,58,62),(40,46,41),(43,59,56),(44,53,61),(48,60,50),(49,54,55),(64,88,68),(65,82,73),(66,76,78),(67,70,83),(69,89,93),(71,77,72),(74,90,87),(75,84,92),(79,91,81),(80,85,86)]])
39 conjugacy classes
| class | 1 | 3A | 3B | 3C | ··· | 3H | 31A | ··· | 31J | 93A | ··· | 93T |
| order | 1 | 3 | 3 | 3 | ··· | 3 | 31 | ··· | 31 | 93 | ··· | 93 |
| size | 1 | 1 | 1 | 31 | ··· | 31 | 3 | ··· | 3 | 3 | ··· | 3 |
39 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 |
| type | + | ||||
| image | C1 | C3 | C3 | C31⋊C3 | C3×C31⋊C3 |
| kernel | C3×C31⋊C3 | C31⋊C3 | C93 | C3 | C1 |
| # reps | 1 | 6 | 2 | 10 | 20 |
Matrix representation of C3×C31⋊C3 ►in GL3(𝔽373) generated by
| 284 | 0 | 0 |
| 0 | 284 | 0 |
| 0 | 0 | 284 |
| 105 | 4 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 284 | 16 | 31 |
| 220 | 292 | 356 |
G:=sub<GL(3,GF(373))| [284,0,0,0,284,0,0,0,284],[105,1,0,4,0,1,1,0,0],[1,284,220,0,16,292,0,31,356] >;
C3×C31⋊C3 in GAP, Magma, Sage, TeX
C_3\times C_{31}\rtimes C_3 % in TeX
G:=Group("C3xC31:C3"); // GroupNames label
G:=SmallGroup(279,3);
// by ID
G=gap.SmallGroup(279,3);
# by ID
G:=PCGroup([3,-3,-3,-31,2027]);
// Polycyclic
G:=Group<a,b,c|a^3=b^31=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
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