Finite Group Databasebeta

← Prev (#4602) Next (#4604) →

Cayley diagram

C3×#356 #4603

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) = C2

Aliases

direct productC3×#356

Subgroups

subgroupG/H
center, socleC2×C3#99
commutatorC2×C4C2×C3×C4
Frattini, cosocleC2×Q8C22×C3
prime radicalC22×C3C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ G

60 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:

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

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67550, #67551, #67552, …
maximal subgroup of#67550, #67551, #67552, …
maximal normal subgroup of#67550, #67551, #67552, …
maximal quotient ofC3×#1319, C3×#1325, C3×#1346, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 162);