Finite Group Databasebeta

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Cayley diagram

C3×#568 #5898

order192
nilpotency class2
isoclinismisoclinic class of H(5,𝔽2)
automorphismAut(G) = C2×#72217027 (order 3072) · Inn(G) = C24 · Out(G) = C23×S4
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25

Aliases

direct productC3×#568

Subgroups

subgroupG/H
centerC3×C4C24
commutatorC2C23×C3×C4
Frattini, cosocleC4C24×C3
socleC2×C3C23×C4
prime radicalC2×C3×#41C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C3×C4 ◁ G

258 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:

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72020, #72021, #72022, …
maximal subgroup of#72020, #72021, #72022, …
maximal normal subgroup of#72020, #72021, #72022, …
maximal quotient ofC3×#2873, C3×#2875, C3×#2879, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1457);