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C2×C3×#2598 #10653464

order768
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C29

Aliases

direct productC2×C3×#2598

Subgroups

subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC23C25×C3
prime radicalC2×C3×#392C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

3502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 384

A minimal faithful permutation representation — 5 generators on 25 points:

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

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72254604, C2×D3×#2598, C2×#11427642, …
maximal subgroup of#72254604, C2×D3×#2598, C3×#619101, …
maximal normal subgroup of#72254604, C2×D3×#2598, C3×#619101, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 27845);