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C3×#92537 #13093747

order1500
isoclinismisoclinic class of #92537
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×#92537

Splittings

Fitting, HallC3×H(3,𝔽5) ⋊ C4
derived, SylowH(3,𝔽5) ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC3#92537
commutator, hypocenterH(3,𝔽5)C3×C4
FrattiniC5C3×#1043
FittingC3×H(3,𝔽5)C4
socleC3×C5#1043
prime radicalC3×#6953C2
cosocle#6953C2×C3
LCSG ▷ H(3,𝔽5)
UCSC1 ◁ C3

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 75

A minimal faithful permutation representation — 3 generators on 28 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#92537⋊C2, C2×C3×#92537
maximal normal subgroup ofC3×#92537⋊C2, C2×C3×#92537
maximal quotient ofC2×C3×#92537

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1500, 48);