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Cayley diagram

C2×D75 #64468

order300
isoclinismisoclinic class of D75
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD150
direct productC2×D75
unambiguous semidirect product(C2×C3)⋊D25
unambiguous semidirect product(C2×C25)⋊D3

Splittings

FittingC2×C3×C25 ⋊ C2
derived, HallC3×C25 ⋊ C22
SylowC25 ⋊ C2×D3
SylowC3 ⋊ C2×D25

Subgroups

subgroupG/H
center, hypercenterC2D75
commutator, cosocle, hypocenterC3×C25C22
FrattiniC5C2×D15
FittingC2×C3×C25C2
socleC2×C3×C5D5
LCSG ▷ C3×C25
UCSC1 ◁ C2

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 150

A minimal faithful permutation representation — 2 generators on 30 points:

(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);(4,5)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10597028, #10597029, #10597032, …
maximal normal subgroup of#10597028, #10597029, #10597032, …
maximal quotient ofDic150, C4×D75, D300, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(300, 11);