Finite Group Databasebeta

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Cayley diagram

C3×#365 #4612

order192
nilpotency class4
isoclinismisoclinic class of #365
automorphismAut(G) = C2×#3109 (order 256) · Inn(G) = #106 · Out(G) = C23
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×#365

Subgroups

subgroupG/H
center, socleC2×C3#106
commutatorC8C2×C3×C4
Frattini, cosocleC2×C8C22×C3
prime radicalC22×C3SD16
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ G

42 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#69309, #69310, D3×#365, …
maximal subgroup of#69309, #69310, D3×#365, …
maximal normal subgroup of#69309, #69310, D3×#365, …
maximal quotient ofC3×#1368, C3×#1383, C2×C3×#365, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 171);