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C3×#11030 #10629637

order768
nilpotency class3
isoclinismisoclinic class of #2008
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26

Aliases

direct productC3×#11030

Subgroups

subgroupG/H
center, socleC22×C3#392
commutatorC22×C4C22×C3×C4
Frattini, cosocleC23×C4C23×C3
prime radicalC22×C3×AGL(1,ℤ/8)C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

1422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 96

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

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72248604, #72248614, D3×#11030, …
maximal subgroup of#72248604, #72248614, C3×#364592, …
maximal normal subgroup of#72248604, #72248614, C3×#364592, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 4018);