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C3×#25771 #10644378

order768
nilpotency class3
isoclinismisoclinic class of #24404
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26×C4

Aliases

direct productC3×#25771

Subgroups

subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

2090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 15 points:

(1,2,3)(5,6)(8,9)(10,12)(11,13)(14,15);(4,5,7,6)(8,10)(9,12)(11,15)(13,14);(8,9)(10,14,15,12)(11,13);(4,6,7,5)

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#25771, #72270983, #72271051, …
maximal subgroup ofD3×#25771, #72270983, C3×D4×B(4,𝔽2), …
maximal normal subgroup ofD3×#25771, #72270983, C3×D4×B(4,𝔽2), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 18759);