Finite Group Databasebeta

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Cayley diagram

C3×#560 #5890

order192
nilpotency class2
isoclinismisoclinic class of #560
automorphismAut(G) = C2×#72209712 (order 3072) · Inn(G) = C24 · Out(G) = C2×#1020
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C24

Aliases

direct productC3×#560

Subgroups

subgroupG/H
center, socleC22×C3C24
commutator, Frattini, cosocleC22C24×C3
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

252 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#71947, #87070, #87071, …
maximal subgroup of#71947, C3×#2200, #87070, …
maximal normal subgroup of#71947, C3×#2200, #87070, …
maximal quotient ofC3×#2375, C3×#2376, C3×#2377, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1449);