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C3×#12575 #10631182

order768
nilpotency class4
isoclinismisoclinic class of #2121
automorphism?
homologyAbelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C23×C4

Aliases

direct productC3×#12575

Subgroups

subgroupG/H
center, socleC22×C3C2×#99
commutatorC23C2×C3×C42
Frattini, cosocleC22×D4C23×C3
prime radicalC22×C3×#41C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×#41 ◁ G

810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72253910, #72253913, D3×#12575, …
maximal subgroup of#72253910, #72253913, C3×#147813, …
maximal normal subgroup of#72253910, #72253913, C3×#147813, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5563);