Finite Group Databasebeta

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Cayley diagram

C3×#361 #4608

order192
nilpotency class4
isoclinismisoclinic class of #360
automorphismAut(G) = C2×#61451 (order 512) · Inn(G) = #102 · Out(G) = C24
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#361

Subgroups

subgroupG/H
center, socleC2×C3#102
commutatorC8C2×C3×C4
Frattini, cosocleC2×C8C22×C3
prime radicalC2×C3×D8C2
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ G

62 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:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#69254, D3×#361, #69256, …
maximal subgroup of#69254, D3×#361, #69256, …
maximal normal subgroup of#69254, D3×#361, #69256, …
maximal quotient ofC3×#1333, C3×#1349, C3×#1361, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 167);