Finite Group Databasebeta

← Prev (#10633876) Next (#10633878) →

C3×#15270 #10633877

order768
nilpotency class2
isoclinismisoclinic class of #15257
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C27

Aliases

direct productC3×#15270

Subgroups

subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC2×C3×#392C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

2466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 384

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

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72255258, #72255451, #72255473, …
maximal subgroup of#72255258, #72255451, #72255473, …
maximal normal subgroup of#72255258, #72255451, #72255473, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 8258);