Finite Group Databasebeta

← Prev (#4601) Next (#4603) →

Cayley diagram

C3×#355 #4602

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×#355

Subgroups

subgroupG/H
center, socleC2×C3#99
commutatorC2×C4C2×C3×C4
Frattini, cosocleC2×Q8C22×C3
prime radicalC2×C3×D4C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ G

64 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:

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

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67546, #67547, #67548, …
maximal subgroup of#67546, #67547, #67548, …
maximal normal subgroup of#67546, #67547, #67548, …
maximal quotient ofC3×#1318, C3×#1324, C3×#1345, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 161);