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Cayley diagram

C3×M32 #853

order96
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C22×#41 (order 64) · Inn(G) = C22 · Out(G) = C24
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C1

Aliases

direct productC3×M32

Subgroups

subgroupG/H
centerC3×C8C22
commutatorC2C2×C3×C8
Frattini, cosocleC8C22×C3
socleC2×C3C2×C8
prime radicalC22×C3C8
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C3×C8 ◁ G

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2,3,7,4,8,9,14,5,6,10,12,11,13,15,16)(17,18,19);(2,6)(7,12)(8,13)(14,16)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#4511, #4512, #4513, …
maximal subgroup of#4511, #4512, #4513, …
maximal normal subgroup of#4511, #4512, #4513, …
maximal quotient ofC3×#346, C3×C22⋊C16, C3×C4⋊C16, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 60);