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Cayley diagram

C3×#99 #842

order96
nilpotency class3
isoclinismisoclinic class of #99
automorphismAut(G) = C2×B(4,𝔽2) (order 128) · Inn(G) = C22⋊C4 · Out(G) = C23
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×#99

Subgroups

subgroupG/H
center, socleC2×C3C22⋊C4
commutatorC22C2×C3×C4
Frattini, cosocleC23C22×C3
prime radicalC2×C3×D4C2
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ G

52 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)(3,7,8,5)(4,6)(9,10,11);(1,3)(2,5)(4,8)(6,7)

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#4471, #4472, #4473, …
maximal subgroup of#4471, #4472, #4473, …
maximal normal subgroup of#4471, #4472, #4473, …
maximal quotient ofC3×#323, C3×#324, C3×#327, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 49);