Finite Group Databasebeta

← Prev (#10629642) Next (#10629644) →

C3×#11036 #10629643

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

Aliases

direct productC3×#11036

Subgroups

subgroupG/H
center, socleC22×C3#392
commutatorC22×C4C22×C3×C4
Frattini, cosocleC23×C4C23×C3
prime radicalC2×C3×#575C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

1310 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)(4,5,6,7)(9,14)(11,12)(15,22)(16,17)(18,23)(19,21);(8,9)(10,18)(11,17)(12,14)(13,16)(15,19)(20,22)(21,23);(8,10)(9,15)(11,19)(12,21)(13,20)(14,18)(16,23)(17,22)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#11036, #72248210, #72271781
maximal subgroup ofD3×#11036, #72248210, C3×#394080, …
maximal normal subgroup ofD3×#11036, #72248210, C3×#394080, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 4024);