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Cayley diagram

C3×#485 #5359

order192
nilpotency class3
isoclinismisoclinic class of #447
automorphismAut(G) = C26×D4 (order 512) · Inn(G) = C2×D4 · Out(G) = C25
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C2

Aliases

direct productC3×#485

Subgroups

subgroupG/H
center, socleC22×C3C2×D4
commutator, Frattini, cosocleC2×C4C23×C3
prime radicalC23×C3D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ 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)(3,6)(4,5)(7,8)(9,10,13,17)(11,14,19,23)(12,15,20,16)(18,21,24,22)(25,26,27);(2,5)(6,8)(9,11,13,19)(10,14,17,23)(12,18,20,24)(15,21,16,22);(1,3,4,7)(2,6,5,8)(10,15)(12,20)(14,21)(16,17)(18,24)(22,23)

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#81492, #81493, #81494, …
maximal subgroup of#81492, #81493, #81494, …
maximal normal subgroup of#81492, #81493, #81494, …
maximal quotient ofC3×#1825, C3×#1854, C3×#1873, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 918);