Finite Group Databasebeta

← Prev (#5870) Next (#5872) →

Cayley diagram

C3×#541 #5871

order192
nilpotency class2
isoclinismisoclinic class of #534
automorphismAut(G) = C2×#72215179 (order 2048) · Inn(G) = C24 · Out(G) = C22×#99
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C24

Aliases

direct productC3×#541

Subgroups

subgroupG/H
center, socleC22×C3C24
commutator, Frattini, cosocleC22C24×C3
prime radicalC23×C3C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 27 points:

(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,27);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24);(2,6)(5,8)(11,18)(14,17)(19,24)(22,23);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#83442, #83443, #83444, …
maximal subgroup of#83442, #83443, #83444, …
maximal normal subgroup of#83442, #83443, #83444, …
maximal quotient ofC3×#2314, C3×#2320, C3×#2323, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1430);