Finite Group Databasebeta

← Prev (#5352) Next (#5354) →

Cayley diagram

C3×#479 #5353

order192
nilpotency class3
isoclinismisoclinic class of #447
automorphismAut(G) = C26×D4 (order 512) · Inn(G) = C2×D4 · Out(G) = C25
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C2

Aliases

direct productC3×#479

Subgroups

subgroupG/H
center, socle, prime radicalC22×C3C2×D4
commutator, Frattini, cosocleC2×C4C23×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ G

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,4,6)(3,7,8,5)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,22,23,19)(25,26,27);(1,3,4,8)(2,5,6,7)(10,14)(15,23)(16,21)(17,24);(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17)

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#81204, #81205, #81206, …
maximal subgroup of#81204, #81205, #81206, …
maximal normal subgroup of#81204, #81205, #81206, …
maximal quotient ofC3×#1805, C3×#1863, C3×#1865, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 912);