Finite Group Databasebeta

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Cayley diagram

C3×#100 #843

order96
nilpotency class3
isoclinismisoclinic class of #99
automorphismAut(G) = C2×#2196 (order 256) · Inn(G) = C22⋊C4 · Out(G) = C2×D4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#100

Subgroups

subgroupG/H
center, socleC2×C3C22⋊C4
commutatorC22C2×C3×C4
Frattini, cosocleC2×C4C22×C3
prime radicalC2×C3×D4C2
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ G

46 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 11 points:

(1,2,3,6,4,5,7,8)(9,10,11);(2,5)(3,7)

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#4475, #4476, D3×#100, …
maximal subgroup of#4475, #4476, C3×AΣL(1,𝔽16), …
maximal normal subgroup of#4475, #4476, C3×AΣL(1,𝔽16), …
maximal quotient ofC3×#323, C3×#329, C3×#331, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 50);