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G = S3×C51order 306 = 2·32·17

Direct product of C51 and S3

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: S3×C51, C3⋊C102, C513C6, C321C34, (C3×C51)⋊4C2, SmallGroup(306,6)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C51
C1C3C51C3×C51 — S3×C51
C3 — S3×C51
C1C51

Generators and relations for S3×C51
 G = < a,b,c | a51=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
2C3
3C6
3C34
2C51
3C102

Smallest permutation representation of S3×C51
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 35 18)(2 36 19)(3 37 20)(4 38 21)(5 39 22)(6 40 23)(7 41 24)(8 42 25)(9 43 26)(10 44 27)(11 45 28)(12 46 29)(13 47 30)(14 48 31)(15 49 32)(16 50 33)(17 51 34)(52 69 86)(53 70 87)(54 71 88)(55 72 89)(56 73 90)(57 74 91)(58 75 92)(59 76 93)(60 77 94)(61 78 95)(62 79 96)(63 80 97)(64 81 98)(65 82 99)(66 83 100)(67 84 101)(68 85 102)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)

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), (1,35,18)(2,36,19)(3,37,20)(4,38,21)(5,39,22)(6,40,23)(7,41,24)(8,42,25)(9,43,26)(10,44,27)(11,45,28)(12,46,29)(13,47,30)(14,48,31)(15,49,32)(16,50,33)(17,51,34)(52,69,86)(53,70,87)(54,71,88)(55,72,89)(56,73,90)(57,74,91)(58,75,92)(59,76,93)(60,77,94)(61,78,95)(62,79,96)(63,80,97)(64,81,98)(65,82,99)(66,83,100)(67,84,101)(68,85,102), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)>;

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), (1,35,18)(2,36,19)(3,37,20)(4,38,21)(5,39,22)(6,40,23)(7,41,24)(8,42,25)(9,43,26)(10,44,27)(11,45,28)(12,46,29)(13,47,30)(14,48,31)(15,49,32)(16,50,33)(17,51,34)(52,69,86)(53,70,87)(54,71,88)(55,72,89)(56,73,90)(57,74,91)(58,75,92)(59,76,93)(60,77,94)(61,78,95)(62,79,96)(63,80,97)(64,81,98)(65,82,99)(66,83,100)(67,84,101)(68,85,102), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72) );

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)], [(1,35,18),(2,36,19),(3,37,20),(4,38,21),(5,39,22),(6,40,23),(7,41,24),(8,42,25),(9,43,26),(10,44,27),(11,45,28),(12,46,29),(13,47,30),(14,48,31),(15,49,32),(16,50,33),(17,51,34),(52,69,86),(53,70,87),(54,71,88),(55,72,89),(56,73,90),(57,74,91),(58,75,92),(59,76,93),(60,77,94),(61,78,95),(62,79,96),(63,80,97),(64,81,98),(65,82,99),(66,83,100),(67,84,101),(68,85,102)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,81),(10,82),(11,83),(12,84),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(27,99),(28,100),(29,101),(30,102),(31,52),(32,53),(33,54),(34,55),(35,56),(36,57),(37,58),(38,59),(39,60),(40,61),(41,62),(42,63),(43,64),(44,65),(45,66),(46,67),(47,68),(48,69),(49,70),(50,71),(51,72)])

153 conjugacy classes

class 1  2 3A3B3C3D3E6A6B17A···17P34A···34P51A···51AF51AG···51CB102A···102AF
order12333336617···1734···3451···5151···51102···102
size1311222331···13···31···12···23···3

153 irreducible representations

dim111111112222
type+++
imageC1C2C3C6C17C34C51C102S3C3×S3S3×C17S3×C51
kernelS3×C51C3×C51S3×C17C51C3×S3C32S3C3C51C17C3C1
# reps112216163232121632

Matrix representation of S3×C51 in GL2(𝔽103) generated by

190
019
,
4699
056
,
8518
4518
G:=sub<GL(2,GF(103))| [19,0,0,19],[46,0,99,56],[85,45,18,18] >;

S3×C51 in GAP, Magma, Sage, TeX

S_3\times C_{51}
% in TeX

G:=Group("S3xC51");
// GroupNames label

G:=SmallGroup(306,6);
// by ID

G=gap.SmallGroup(306,6);
# by ID

G:=PCGroup([4,-2,-3,-17,-3,3267]);
// Polycyclic

G:=Group<a,b,c|a^51=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of S3×C51 in TeX

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