direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C34, C6⋊C34, C102⋊3C2, C51⋊4C22, C3⋊(C2×C34), SmallGroup(204,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C34 |
Generators and relations for S3×C34
G = < a,b,c | a34=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C34
►On 102 points
Generators in S102
(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)
(1 62 74)(2 63 75)(3 64 76)(4 65 77)(5 66 78)(6 67 79)(7 68 80)(8 35 81)(9 36 82)(10 37 83)(11 38 84)(12 39 85)(13 40 86)(14 41 87)(15 42 88)(16 43 89)(17 44 90)(18 45 91)(19 46 92)(20 47 93)(21 48 94)(22 49 95)(23 50 96)(24 51 97)(25 52 98)(26 53 99)(27 54 100)(28 55 101)(29 56 102)(30 57 69)(31 58 70)(32 59 71)(33 60 72)(34 61 73)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(35 98)(36 99)(37 100)(38 101)(39 102)(40 69)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 81)(53 82)(54 83)(55 84)(56 85)(57 86)(58 87)(59 88)(60 89)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 97)
G:=sub<Sym(102)| (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), (1,62,74)(2,63,75)(3,64,76)(4,65,77)(5,66,78)(6,67,79)(7,68,80)(8,35,81)(9,36,82)(10,37,83)(11,38,84)(12,39,85)(13,40,86)(14,41,87)(15,42,88)(16,43,89)(17,44,90)(18,45,91)(19,46,92)(20,47,93)(21,48,94)(22,49,95)(23,50,96)(24,51,97)(25,52,98)(26,53,99)(27,54,100)(28,55,101)(29,56,102)(30,57,69)(31,58,70)(32,59,71)(33,60,72)(34,61,73), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(35,98)(36,99)(37,100)(38,101)(39,102)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,85)(57,86)(58,87)(59,88)(60,89)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,97)>;
G:=Group( (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), (1,62,74)(2,63,75)(3,64,76)(4,65,77)(5,66,78)(6,67,79)(7,68,80)(8,35,81)(9,36,82)(10,37,83)(11,38,84)(12,39,85)(13,40,86)(14,41,87)(15,42,88)(16,43,89)(17,44,90)(18,45,91)(19,46,92)(20,47,93)(21,48,94)(22,49,95)(23,50,96)(24,51,97)(25,52,98)(26,53,99)(27,54,100)(28,55,101)(29,56,102)(30,57,69)(31,58,70)(32,59,71)(33,60,72)(34,61,73), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(35,98)(36,99)(37,100)(38,101)(39,102)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,85)(57,86)(58,87)(59,88)(60,89)(61,90)(62,91)(63,92)(64,93)(65,94)(66,95)(67,96)(68,97) );
G=PermutationGroup([[(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)], [(1,62,74),(2,63,75),(3,64,76),(4,65,77),(5,66,78),(6,67,79),(7,68,80),(8,35,81),(9,36,82),(10,37,83),(11,38,84),(12,39,85),(13,40,86),(14,41,87),(15,42,88),(16,43,89),(17,44,90),(18,45,91),(19,46,92),(20,47,93),(21,48,94),(22,49,95),(23,50,96),(24,51,97),(25,52,98),(26,53,99),(27,54,100),(28,55,101),(29,56,102),(30,57,69),(31,58,70),(32,59,71),(33,60,72),(34,61,73)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(35,98),(36,99),(37,100),(38,101),(39,102),(40,69),(41,70),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,78),(50,79),(51,80),(52,81),(53,82),(54,83),(55,84),(56,85),(57,86),(58,87),(59,88),(60,89),(61,90),(62,91),(63,92),(64,93),(65,94),(66,95),(67,96),(68,97)]])
S3×C34 is a maximal subgroup of
C51⋊D4 C17⋊D12
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 17A | ··· | 17P | 34A | ··· | 34P | 34Q | ··· | 34AV | 51A | ··· | 51P | 102A | ··· | 102P |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C17 | C34 | C34 | S3 | D6 | S3×C17 | S3×C34 |
| kernel | S3×C34 | S3×C17 | C102 | D6 | S3 | C6 | C34 | C17 | C2 | C1 |
| # reps | 1 | 2 | 1 | 16 | 32 | 16 | 1 | 1 | 16 | 16 |
Matrix representation of S3×C34 ►in GL2(𝔽103) generated by
| 42 | 0 |
| 0 | 42 |
| 0 | 102 |
| 1 | 102 |
| 102 | 1 |
| 0 | 1 |
G:=sub<GL(2,GF(103))| [42,0,0,42],[0,1,102,102],[102,0,1,1] >;
S3×C34 in GAP, Magma, Sage, TeX
S_3\times C_{34} % in TeX
G:=Group("S3xC34"); // GroupNames label
G:=SmallGroup(204,10);
// by ID
G=gap.SmallGroup(204,10);
# by ID
G:=PCGroup([4,-2,-2,-17,-3,2179]);
// Polycyclic
G:=Group<a,b,c|a^34=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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