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Cayley diagram

C3×#510 #5384

order192
nilpotency class4
isoclinismisoclinic class of #509
automorphismAut(G) = C2×#61451 (order 512) · Inn(G) = C2×D8 · Out(G) = C24
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC3×#510

Subgroups

subgroupG/H
center, socleC2×C3C2×D8
commutator, Frattini, cosocleC8C23×C3
prime radicalC3×#135C2
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ G

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2)(3,11)(4,15)(5,20)(6,8)(7,12)(9,18)(10,17)(13,31)(14,28)(16,26)(19,27)(21,25)(22,30)(23,32)(24,29)(33,34,35);(1,3,6,14)(2,7,8,22)(4,16,19,32)(5,21,17,13)(9,25,24,31)(10,26,20,23)(11,27,28,15)(12,29,30,18);(2,8)(7,22)(9,24)(10,20)(11,28)(15,27)(23,26)(25,31)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#71519, #71520, #71521, …
maximal subgroup of#71519, #71520, #71521, …
maximal normal subgroup of#71519, #71520, #71521, …
maximal quotient ofC3×#2139, C3×#2141, C3×#2159, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 943);