Finite Group Databasebeta

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Cayley diagram

C3×#555 #5885

order192
nilpotency class2
isoclinismisoclinic class of D42
automorphismAut(G) = C2×#72215688 (order 4096) · Inn(G) = C24 · Out(G) = C2×#2196
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4

Aliases

direct productC3×#555

Subgroups

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

236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#71927, #71928, #71929, …
maximal subgroup of#71927, #71928, #71929, …
maximal normal subgroup of#71927, #71928, #71929, …
maximal quotient ofC3×#2346, C3×#2354, C3×#2363, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1444);