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Cayley diagram

C3×#103 #846

order96
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C24×D4 (order 128) · Inn(G) = D4 · Out(G) = C24
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#103

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

42 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);(5,6,8,10)(7,12,11,9)

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#4788, #4789, #4790, …
maximal subgroup of#4788, #4789, #4790, …
maximal normal subgroup of#4788, #4789, #4790, …
maximal quotient ofC3×Q8⋊C8, C3×#328, C3×#332, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 53);