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Cayley diagram

C3×#578 #5908

order192
nilpotency class3
isoclinismisoclinic class of #576
automorphismAut(G) = C2×S4×AGL(1,ℤ/8) (order 1536) · Inn(G) = C22×D4 · Out(G) = C23×D3
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productC3×#578

Subgroups

subgroupG/H
center, socleC2×C3C22×D4
commutator, Frattini, cosocleC4C24×C3
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C3×#41 ◁ G

248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 35 points:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72124, #72125, #84016, …
maximal subgroup of#72124, #72125, C3×#2193, …
maximal normal subgroup of#72124, #72125, C3×#2193, …
maximal quotient ofC3×#2898, C3×#2913, C3×#2947, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1467);