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C3×#87739 #11924900

order1200
isoclinismisoclinic class of #87735
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×#87739

Splittings

HallC3×C25 ⋊ M16
SylowC25 ⋊ C3×M16

Subgroups

subgroupG/H
centerC2×C3C2×#1035
commutatorC2×C25C2×C3×C4
FrattiniC2×C5C2×C3×AGL(1,𝔽5)
FittingC22×C3×C25C4
socleC2×C3×C5C2×AGL(1,𝔽5)
prime radicalC22×C3×C5AGL(1,𝔽5)
cosocleDic25C22×C3
hypercenterC22×C3#1035
hypocenterC25C3×M16
LCSG ▷ C2×C25 ▷ C25
UCSC1 ◁ C2×C3 ◁ C22×C3

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 600

A minimal faithful permutation representation — 2 generators on 36 points:

(1,2,3)(4,10,15,13,8,6,22,20,18,16,11,28,27,19,25,23,24,9,14,17,12,5,7,26,21)(30,33)(34,36);(5,7,8,13)(6,11,12,23)(9,17,18,22)(10,20,21,14)(15,25,26,28)(16,19,24,27)(29,30,31,34,32,33,35,36)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1200, 110);