direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C32, C96⋊5C2, D6.2C16, C16.19D6, C48.24C22, Dic3.2C16, C3⋊C32⋊6C2, C3⋊1(C2×C32), C3⋊C8.4C8, C3⋊C16.3C4, (C4×S3).4C8, (S3×C8).5C4, C4.16(S3×C8), C8.36(C4×S3), C2.1(S3×C16), C6.1(C2×C16), (S3×C16).3C2, C24.57(C2×C4), C12.21(C2×C8), SmallGroup(192,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C32 |
Generators and relations for S3×C32
G = < a,b,c | a32=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C32
►On 96 points
Generators in S96
(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)
(1 94 48)(2 95 49)(3 96 50)(4 65 51)(5 66 52)(6 67 53)(7 68 54)(8 69 55)(9 70 56)(10 71 57)(11 72 58)(12 73 59)(13 74 60)(14 75 61)(15 76 62)(16 77 63)(17 78 64)(18 79 33)(19 80 34)(20 81 35)(21 82 36)(22 83 37)(23 84 38)(24 85 39)(25 86 40)(26 87 41)(27 88 42)(28 89 43)(29 90 44)(30 91 45)(31 92 46)(32 93 47)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(33 95)(34 96)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)
G:=sub<Sym(96)| (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), (1,94,48)(2,95,49)(3,96,50)(4,65,51)(5,66,52)(6,67,53)(7,68,54)(8,69,55)(9,70,56)(10,71,57)(11,72,58)(12,73,59)(13,74,60)(14,75,61)(15,76,62)(16,77,63)(17,78,64)(18,79,33)(19,80,34)(20,81,35)(21,82,36)(22,83,37)(23,84,38)(24,85,39)(25,86,40)(26,87,41)(27,88,42)(28,89,43)(29,90,44)(30,91,45)(31,92,46)(32,93,47), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(33,95)(34,96)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)>;
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), (1,94,48)(2,95,49)(3,96,50)(4,65,51)(5,66,52)(6,67,53)(7,68,54)(8,69,55)(9,70,56)(10,71,57)(11,72,58)(12,73,59)(13,74,60)(14,75,61)(15,76,62)(16,77,63)(17,78,64)(18,79,33)(19,80,34)(20,81,35)(21,82,36)(22,83,37)(23,84,38)(24,85,39)(25,86,40)(26,87,41)(27,88,42)(28,89,43)(29,90,44)(30,91,45)(31,92,46)(32,93,47), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(33,95)(34,96)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94) );
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)], [(1,94,48),(2,95,49),(3,96,50),(4,65,51),(5,66,52),(6,67,53),(7,68,54),(8,69,55),(9,70,56),(10,71,57),(11,72,58),(12,73,59),(13,74,60),(14,75,61),(15,76,62),(16,77,63),(17,78,64),(18,79,33),(19,80,34),(20,81,35),(21,82,36),(22,83,37),(23,84,38),(24,85,39),(25,86,40),(26,87,41),(27,88,42),(28,89,43),(29,90,44),(30,91,45),(31,92,46),(32,93,47)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(33,95),(34,96),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,91),(62,92),(63,93),(64,94)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 16A | ··· | 16H | 16I | ··· | 16P | 24A | 24B | 24C | 24D | 32A | ··· | 32P | 32Q | ··· | 32AF | 48A | ··· | 48H | 96A | ··· | 96P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 16 | ··· | 16 | 16 | ··· | 16 | 24 | 24 | 24 | 24 | 32 | ··· | 32 | 32 | ··· | 32 | 48 | ··· | 48 | 96 | ··· | 96 |
| size | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 3 | 3 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | C32 | S3 | D6 | C4×S3 | S3×C8 | S3×C16 | S3×C32 |
| kernel | S3×C32 | C3⋊C32 | C96 | S3×C16 | C3⋊C16 | S3×C8 | C3⋊C8 | C4×S3 | Dic3 | D6 | S3 | C32 | C16 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 32 | 1 | 1 | 2 | 4 | 8 | 16 |
Matrix representation of S3×C32 ►in GL2(𝔽97) generated by
| 46 | 0 |
| 0 | 46 |
| 0 | 96 |
| 1 | 96 |
| 0 | 96 |
| 96 | 0 |
G:=sub<GL(2,GF(97))| [46,0,0,46],[0,1,96,96],[0,96,96,0] >;
S3×C32 in GAP, Magma, Sage, TeX
S_3\times C_{32} % in TeX
G:=Group("S3xC32"); // GroupNames label
G:=SmallGroup(192,5);
// by ID
G=gap.SmallGroup(192,5);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,36,58,80,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^32=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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