Finite Group Databasebeta

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Cayley diagram

C3×#354 #4601

order192
nilpotency class4
isoclinismisoclinic class of #353
automorphismAut(G) = C2×#33551 (order 512) · Inn(G) = #99 · Out(G) = C24
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×#354

Subgroups

subgroupG/H
center, socleC2×C3#99
commutatorC2×C4C2×C3×C4
Frattini, cosocleC2×D4C22×C3
prime radicalC2×C3×D4C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ G

70 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)(3,9,11,16)(4,12,14,7)(5,8)(6,10)(13,15)(17,18,19);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67542, #67543, #67544, …
maximal subgroup of#67542, #67543, #67544, …
maximal normal subgroup of#67542, #67543, #67544, …
maximal quotient ofC3×#1319, C3×#1324, C3×#1344, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 160);