Finite Group Databasebeta

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Cayley diagram

C3×#552 #5882

order192
nilpotency class2
isoclinismisoclinic class of D42
automorphismAut(G) = C25×#120 (order 1024) · Inn(G) = C24 · Out(G) = C23×D4
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C24

Aliases

direct productC3×#552

Subgroups

subgroupG/H
center, socleC22×C3C24
commutator, Frattini, cosocleC22C24×C3
prime radicalC2×C3×D4C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

214 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)(9,10,13,12)(11,15,16,14)(17,18,19);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,15)(13,16);(2,6)(5,8)(9,12,13,10)(11,15,16,14);(11,16)(14,15)

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#83590, #83591, #83592, …
maximal subgroup of#83590, #83591, #83592, …
maximal normal subgroup of#83590, #83591, #83592, …
maximal quotient ofC3×#2343, C3×#2344, C3×#2346, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1441);