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C2×D135 #10587351

order540
isoclinismisoclinic class of D135
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD270
direct productC2×D135
unambiguous semidirect product(C2×C5)⋊D27
unambiguous semidirect product(C2×C27)⋊D5

Splittings

FittingC2×C5×C27 ⋊ C2
derived, HallC5×C27 ⋊ C22
SylowC27 ⋊ C2×D5
SylowC5 ⋊ C2×D27

Subgroups

subgroupG/H
center, hypercenterC2D135
commutator, cosocle, hypocenterC5×C27C22
FrattiniC9C2×D15
FittingC2×C5×C27C2
socleC2×C3×C5D9
LCSG ▷ C5×C27
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 270

A minimal faithful permutation representation — 2 generators on 34 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)(30,31,32,33,34);(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(31,34)(32,33)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11761663, #11761664, #11761667, …
maximal normal subgroup of#11761663, #11761664, #11761667, …
maximal quotient ofDic270, C4×D135, D540, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(540, 11);