Copied to
clipboard

G = C3×D25order 150 = 2·3·52

Direct product of C3 and D25

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

Aliases: C3×D25, C25⋊C6, C752C2, C15.2D5, C5.(C3×D5), SmallGroup(150,2)

Series: Derived Chief Lower central Upper central

C1C25 — C3×D25
C1C5C25C75 — C3×D25
C25 — C3×D25
C1C3

Generators and relations for C3×D25
 G = < a,b,c | a3=b25=c2=1, ab=ba, ac=ca, cbc=b-1 >

25C2
25C6
5D5
5C3×D5

Smallest permutation representation of C3×D25
On 75 points
Generators in S75
(1 64 26)(2 65 27)(3 66 28)(4 67 29)(5 68 30)(6 69 31)(7 70 32)(8 71 33)(9 72 34)(10 73 35)(11 74 36)(12 75 37)(13 51 38)(14 52 39)(15 53 40)(16 54 41)(17 55 42)(18 56 43)(19 57 44)(20 58 45)(21 59 46)(22 60 47)(23 61 48)(24 62 49)(25 63 50)
(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 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 50)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(52 75)(53 74)(54 73)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)

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

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

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

C3×D25 is a maximal subgroup of   C75⋊C4

42 conjugacy classes

class 1  2 3A3B5A5B6A6B15A15B15C15D25A···25J75A···75T
order123355661515151525···2575···75
size1251122252522222···22···2

42 irreducible representations

dim11112222
type++++
imageC1C2C3C6D5C3×D5D25C3×D25
kernelC3×D25C75D25C25C15C5C3C1
# reps1122241020

Matrix representation of C3×D25 in GL2(𝔽151) generated by

320
032
,
7136
115120
,
7136
1180
G:=sub<GL(2,GF(151))| [32,0,0,32],[71,115,36,120],[71,11,36,80] >;

C3×D25 in GAP, Magma, Sage, TeX

C_3\times D_{25}
% in TeX

G:=Group("C3xD25");
// GroupNames label

G:=SmallGroup(150,2);
// by ID

G=gap.SmallGroup(150,2);
# by ID

G:=PCGroup([4,-2,-3,-5,-5,650,250,1923]);
// Polycyclic

G:=Group<a,b,c|a^3=b^25=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D25 in TeX

׿
×
𝔽