Finite Group Databasebeta

← Prev (#10639222) Next (#10639224) →

C3×#20616 #10639223

order768
nilpotency class3
isoclinismisoclinic class of #3019
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C28

Aliases

direct productC3×#20616

Subgroups

subgroupG/H
center, socleC22×C3C2×#120
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×H(5,𝔽2)C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

3986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72269921, #72269935, #72270093, …
maximal subgroup of#72269921, #72269935, C3×#6628867, …
maximal normal subgroup of#72269921, #72269935, C3×#6628867, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 13604);