Finite Group Databasebeta

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Cayley diagram

C3×#427 #5301

order192
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C2×#72214076 (order 2048) · Inn(G) = D4 · Out(G) = C2×#2403
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4

Aliases

direct productC3×#427

Subgroups

subgroupG/H
centerC2×C3×C4D4
commutatorC4C22×C3×C4
Frattini, cosocleC2×C4C23×C3
socleC22×C3C2×D4
prime radicalC23×C3D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ G

114 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)(12,17)(16,18)(19,20)(21,22,23);(5,7,8,15,9,16,17,20)(6,10,12,14,13,19,11,18);(5,8,9,17)(6,11,13,12)(7,15,16,20)(10,18,19,14)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67724, #67725, #67726, …
maximal subgroup of#67724, #67725, #67726, …
maximal normal subgroup of#67724, #67725, #67726, …
maximal quotient ofC3×#1564, C3×#1568, C3×#1572, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 860);