Finite Group Databasebeta

← Prev (#5319) Next (#5321) →

Cayley diagram

C3×#446 #5320

order192
nilpotency class2
isoclinismisoclinic class of #120
automorphismAut(G) = C24×#120 (order 512) · Inn(G) = C23 · Out(G) = C26
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#446

Subgroups

subgroupG/H
centerC2×C3×C4C23
commutatorC22C22×C3×C4
Frattini, cosocleC2×C4C23×C3
socle, prime radicalC22×C3C22×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ G

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67769, #70950, #70951, …
maximal subgroup of#67769, C3×#1338, #70950, …
maximal normal subgroup of#67769, C3×#1338, #70950, …
maximal quotient ofC3×#1770, C3×#1771, C3×#1916, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 879);