Finite Group Databasebeta

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Cayley diagram

C3×#488 #5362

order192
nilpotency class3
isoclinismisoclinic class of #465
automorphismAut(G) = C2×#392109 (order 1024) · Inn(G) = C2×D4 · Out(G) = C2×#120
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C4

Aliases

direct productC3×#488

Subgroups

subgroupG/H
center, socleC22×C3C2×D4
commutator, Frattini, cosocleC2×C4C23×C3
prime radicalC2×C3×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ G

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 96

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

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67782, #81628, #81629, …
maximal subgroup of#67782, C3×#1414, #81628, …
maximal normal subgroup of#67782, C3×#1414, #81628, …
maximal quotient ofC3×#1835, C3×#1933, C3×#1934, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 921);