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Cayley diagram

C3×#106 #849

order96
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C2×#3003 (order 256) · Inn(G) = D4 · Out(G) = C22×D4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#106

Subgroups

subgroupG/H
center, socle, prime radicalC22×C3D4
commutatorC4C2×C3×C4
Frattini, cosocleC2×C4C22×C3
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ G

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 15 points:

(1,2,3,4)(6,9)(7,11)(10,12)(13,14,15);(5,6,7,9,8,10,11,12)

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#4487, #4488, #4852, …
maximal subgroup of#4487, #4488, C3×#360, …
maximal normal subgroup of#4487, #4488, C3×#360, …
maximal quotient ofC3×#334, C3×#340, C3×#365, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 56);