Finite Group Databasebeta

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Cayley diagram

C3×#419 #5293

order192
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphismAut(G) = C2×#72216616 (order 2048) · Inn(G) = C2×D4 · Out(G) = C22×#120
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC3×#419

Subgroups

subgroupG/H
center, socleC22×C3C2×D4
commutatorC4C22×C3×C4
Frattini, cosocleC2×C4C23×C3
prime radicalC23×C3D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G

150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#70333, #70334, #70335, …
maximal subgroup of#70333, #70334, #70335, …
maximal normal subgroup of#70333, #70334, #70335, …
maximal quotient ofC3×#1479, C3×#1485, C3×#1491, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 852);