direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×D31, D93⋊C2, C31⋊1D6, C3⋊1D62, C93⋊C22, (S3×C31)⋊C2, (C3×D31)⋊C2, SmallGroup(372,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C93 — S3×D31 |
Generators and relations for S3×D31
G = < a,b,c,d | a3=b2=c31=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of S3×D31
►On 93 points
Generators in S93
(1 51 82)(2 52 83)(3 53 84)(4 54 85)(5 55 86)(6 56 87)(7 57 88)(8 58 89)(9 59 90)(10 60 91)(11 61 92)(12 62 93)(13 32 63)(14 33 64)(15 34 65)(16 35 66)(17 36 67)(18 37 68)(19 38 69)(20 39 70)(21 40 71)(22 41 72)(23 42 73)(24 43 74)(25 44 75)(26 45 76)(27 46 77)(28 47 78)(29 48 79)(30 49 80)(31 50 81)
(32 63)(33 64)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 72)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 80)(50 81)(51 82)(52 83)(53 84)(54 85)(55 86)(56 87)(57 88)(58 89)(59 90)(60 91)(61 92)(62 93)
(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 38)(33 37)(34 36)(39 62)(40 61)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(63 69)(64 68)(65 67)(70 93)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 84)(80 83)(81 82)
G:=sub<Sym(93)| (1,51,82)(2,52,83)(3,53,84)(4,54,85)(5,55,86)(6,56,87)(7,57,88)(8,58,89)(9,59,90)(10,60,91)(11,61,92)(12,62,93)(13,32,63)(14,33,64)(15,34,65)(16,35,66)(17,36,67)(18,37,68)(19,38,69)(20,39,70)(21,40,71)(22,41,72)(23,42,73)(24,43,74)(25,44,75)(26,45,76)(27,46,77)(28,47,78)(29,48,79)(30,49,80)(31,50,81), (32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,91)(61,92)(62,93), (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)(32,38)(33,37)(34,36)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(63,69)(64,68)(65,67)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)>;
G:=Group( (1,51,82)(2,52,83)(3,53,84)(4,54,85)(5,55,86)(6,56,87)(7,57,88)(8,58,89)(9,59,90)(10,60,91)(11,61,92)(12,62,93)(13,32,63)(14,33,64)(15,34,65)(16,35,66)(17,36,67)(18,37,68)(19,38,69)(20,39,70)(21,40,71)(22,41,72)(23,42,73)(24,43,74)(25,44,75)(26,45,76)(27,46,77)(28,47,78)(29,48,79)(30,49,80)(31,50,81), (32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,91)(61,92)(62,93), (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)(32,38)(33,37)(34,36)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(63,69)(64,68)(65,67)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82) );
G=PermutationGroup([[(1,51,82),(2,52,83),(3,53,84),(4,54,85),(5,55,86),(6,56,87),(7,57,88),(8,58,89),(9,59,90),(10,60,91),(11,61,92),(12,62,93),(13,32,63),(14,33,64),(15,34,65),(16,35,66),(17,36,67),(18,37,68),(19,38,69),(20,39,70),(21,40,71),(22,41,72),(23,42,73),(24,43,74),(25,44,75),(26,45,76),(27,46,77),(28,47,78),(29,48,79),(30,49,80),(31,50,81)], [(32,63),(33,64),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,71),(41,72),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79),(49,80),(50,81),(51,82),(52,83),(53,84),(54,85),(55,86),(56,87),(57,88),(58,89),(59,90),(60,91),(61,92),(62,93)], [(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),(32,38),(33,37),(34,36),(39,62),(40,61),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(63,69),(64,68),(65,67),(70,93),(71,92),(72,91),(73,90),(74,89),(75,88),(76,87),(77,86),(78,85),(79,84),(80,83),(81,82)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 31A | ··· | 31O | 62A | ··· | 62O | 93A | ··· | 93O |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 31 | ··· | 31 | 62 | ··· | 62 | 93 | ··· | 93 |
| size | 1 | 3 | 31 | 93 | 2 | 62 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D6 | D31 | D62 | S3×D31 |
| kernel | S3×D31 | S3×C31 | C3×D31 | D93 | D31 | C31 | S3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 15 | 15 | 15 |
Matrix representation of S3×D31 ►in GL4(𝔽373) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 73 |
| 0 | 0 | 235 | 371 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 372 | 300 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 372 | 27 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(373))| [1,0,0,0,0,1,0,0,0,0,1,235,0,0,73,371],[1,0,0,0,0,1,0,0,0,0,372,0,0,0,300,1],[0,372,0,0,1,27,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;
S3×D31 in GAP, Magma, Sage, TeX
S_3\times D_{31} % in TeX
G:=Group("S3xD31"); // GroupNames label
G:=SmallGroup(372,8);
// by ID
G=gap.SmallGroup(372,8);
# by ID
G:=PCGroup([4,-2,-2,-3,-31,54,5763]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^31=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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