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G = S3×C25order 150 = 2·3·52

Direct product of C25 and S3

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

Aliases: S3×C25, C3⋊C50, C753C2, C15.C10, C5.(C5×S3), (C5×S3).C5, SmallGroup(150,1)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C25
C1C3C15C75 — S3×C25
C3 — S3×C25
C1C25

Generators and relations for S3×C25
 G = < a,b,c | a25=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C10
3C50

Smallest permutation representation of S3×C25
On 75 points
Generators in S75
(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)
(1 55 48)(2 56 49)(3 57 50)(4 58 26)(5 59 27)(6 60 28)(7 61 29)(8 62 30)(9 63 31)(10 64 32)(11 65 33)(12 66 34)(13 67 35)(14 68 36)(15 69 37)(16 70 38)(17 71 39)(18 72 40)(19 73 41)(20 74 42)(21 75 43)(22 51 44)(23 52 45)(24 53 46)(25 54 47)
(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 73)(42 74)(43 75)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(50 57)

G:=sub<Sym(75)| (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), (1,55,48)(2,56,49)(3,57,50)(4,58,26)(5,59,27)(6,60,28)(7,61,29)(8,62,30)(9,63,31)(10,64,32)(11,65,33)(12,66,34)(13,67,35)(14,68,36)(15,69,37)(16,70,38)(17,71,39)(18,72,40)(19,73,41)(20,74,42)(21,75,43)(22,51,44)(23,52,45)(24,53,46)(25,54,47), (26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,73)(42,74)(43,75)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(50,57)>;

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), (1,55,48)(2,56,49)(3,57,50)(4,58,26)(5,59,27)(6,60,28)(7,61,29)(8,62,30)(9,63,31)(10,64,32)(11,65,33)(12,66,34)(13,67,35)(14,68,36)(15,69,37)(16,70,38)(17,71,39)(18,72,40)(19,73,41)(20,74,42)(21,75,43)(22,51,44)(23,52,45)(24,53,46)(25,54,47), (26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,73)(42,74)(43,75)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(50,57) );

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)], [(1,55,48),(2,56,49),(3,57,50),(4,58,26),(5,59,27),(6,60,28),(7,61,29),(8,62,30),(9,63,31),(10,64,32),(11,65,33),(12,66,34),(13,67,35),(14,68,36),(15,69,37),(16,70,38),(17,71,39),(18,72,40),(19,73,41),(20,74,42),(21,75,43),(22,51,44),(23,52,45),(24,53,46),(25,54,47)], [(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72),(41,73),(42,74),(43,75),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(50,57)])

75 conjugacy classes

class 1  2  3 5A5B5C5D10A10B10C10D15A15B15C15D25A···25T50A···50T75A···75T
order1235555101010101515151525···2550···5075···75
size1321111333322221···13···32···2

75 irreducible representations

dim111111222
type+++
imageC1C2C5C10C25C50S3C5×S3S3×C25
kernelS3×C25C75C5×S3C15S3C3C25C5C1
# reps114420201420

Matrix representation of S3×C25 in GL2(𝔽151) generated by

1250
0125
,
150150
10
,
01
10
G:=sub<GL(2,GF(151))| [125,0,0,125],[150,1,150,0],[0,1,1,0] >;

S3×C25 in GAP, Magma, Sage, TeX

S_3\times C_{25}
% in TeX

G:=Group("S3xC25");
// GroupNames label

G:=SmallGroup(150,1);
// by ID

G=gap.SmallGroup(150,1);
# by ID

G:=PCGroup([4,-2,-5,-5,-3,45,1603]);
// Polycyclic

G:=Group<a,b,c|a^25=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×C25 in TeX

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