Copied to
clipboard

G = S3×C34order 204 = 22·3·17

Direct product of C34 and S3

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S3×C34, C6⋊C34, C1023C2, C514C22, C3⋊(C2×C34), SmallGroup(204,10)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C34
C1C3C51S3×C17 — S3×C34
C3 — S3×C34
C1C34

Generators and relations for S3×C34
 G = < a,b,c | a34=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C22
3C34
3C34
3C2×C34

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

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

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

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

S3×C34 is a maximal subgroup of   C51⋊D4  C17⋊D12

102 conjugacy classes

class 1 2A2B2C 3  6 17A···17P34A···34P34Q···34AV51A···51P102A···102P
order12223617···1734···3434···3451···51102···102
size1133221···11···13···32···22···2

102 irreducible representations

dim1111112222
type+++++
imageC1C2C2C17C34C34S3D6S3×C17S3×C34
kernelS3×C34S3×C17C102D6S3C6C34C17C2C1
# reps121163216111616

Matrix representation of S3×C34 in GL2(𝔽103) generated by

420
042
,
0102
1102
,
1021
01
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

Export

Subgroup lattice of S3×C34 in TeX

׿
×
𝔽