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Cayley diagram

C3×#473 #5347

order192
nilpotency class3
isoclinismisoclinic class of #471
automorphismAut(G) = C2×D4×AGL(1,ℤ/8) (order 512) · Inn(G) = #121 · Out(G) = C24
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC3×#473

Subgroups

subgroupG/H
center, socleC2×C3#121
commutator, Frattini, cosocleC2×C4C23×C3
prime radicalC3×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ G

100 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:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#71096, #71097, #71098, …
maximal subgroup of#71096, #71097, #71098, …
maximal normal subgroup of#71096, #71097, #71098, …
maximal quotient ofC3×#1669, C3×#1670, C3×#1673, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 906);