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Cayley diagram

C3×#117 #957

order96
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×#23900 (order 512) · Inn(G) = C22 · Out(G) = C2×D42
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4

Aliases

direct productC3×#117

Subgroups

subgroupG/H
centerC2×C3×C4C22
commutatorC2C22×C3×C4
Frattini, cosocleC22C23×C3
socleC22×C3C23
prime radicalC23×C3C22
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ G

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#4531, #4532, #4533, …
maximal subgroup of#4531, #4532, #4533, …
maximal normal subgroup of#4531, #4532, #4533, …
maximal quotient ofC3×#376, C3×C4×C22⋊C4, C3×C4×C4⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 164);