Finite Group Databasebeta

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Cayley diagram

C3×#433 #5307

order192
nilpotency class2
isoclinismisoclinic class of #120
automorphismAut(G) = C2×#14679 (order 512) · Inn(G) = C23 · Out(G) = C22×C22⋊C4
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C4

Aliases

direct productC3×#433

Subgroups

subgroupG/H
centerC2×C3×C4C23
commutatorC22C22×C3×C4
Frattini, cosocleC2×C4C23×C3
socleC22×C3C22×C4
prime radicalC23×C3C2×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ G

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#79359, #79360, #79361, …
maximal subgroup of#79359, #79360, #79361, …
maximal normal subgroup of#79359, #79360, #79361, …
maximal quotient ofC3×#1746, C3×#1750, C3×#1768, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 866);