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Cayley diagram

C3×#491 #5365

order192
nilpotency class3
isoclinismisoclinic class of #468
automorphismAut(G) = C2×#72213256 (order 2048) · Inn(G) = C2×D4 · Out(G) = C2×B(4,𝔽2)
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C4

Aliases

direct productC3×#491

Subgroups

subgroupG/H
center, socle, prime radicalC22×C3C2×D4
commutator, Frattini, cosocleC2×C4C23×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ G

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2)(3,5)(4,6)(7,8)(9,10,15,20)(11,21,23,17)(12,24,22,16)(13,19,14,18)(25,26,27);(1,3,4,7)(2,5,6,8)(9,11,13,22,15,23,14,12)(10,16,18,21,20,24,19,17);(2,6)(5,8)(9,12,13,11,15,22,14,23)(10,17,18,16,20,21,19,24)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67785, #67786, #71251, …
maximal subgroup of#67785, #67786, C3×#1356, …
maximal normal subgroup of#67785, #67786, C3×#1356, …
maximal quotient ofC3×#1836, C3×#1935, C3×#1960, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 924);