Finite Group Databasebeta

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Cayley diagram

C3×#1181 #66826

order348
isoclinismisoclinic class of #1181
automorphismAut(G) = C2×AGL(1,𝔽29) (order 1624) · Inn(G) = #1181 · Out(G) = C2×C7
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×#1181

Splittings

Fitting, socle, HallC3×C29 ⋊ C4
derived, SylowC29 ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC3#1181
commutator, hypocenterC29C3×C4
FrattiniC1G
Fitting, socleC3×C29C4
prime radicalC3×D29C2
cosocleD29C2×C3
LCSG ▷ C29
UCSC1 ◁ C3

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 87

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#1181, C2×C3×#1181, C9×#1181, …
maximal normal subgroup ofD3×#1181, C2×C3×#1181, C9×#1181, …
maximal quotient ofC3×#6599, C2×C3×#1181, C9×#1181, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(348, 5);