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Cayley diagram

C2×#41⋊C3 #993

order96
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = D4×S4 (order 192) · Inn(G) = A4 · Out(G) = C2×D4
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×#41⋊C3
unambiguous semidirect product(C2×#41)⋊C3

Splittings

Fitting, SylowC2×#41 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2×C4A4
commutator, cosocle, hypocenterQ8C22×C3
FrattiniC2C22×A4
FittingC2×#41C3
socleC22C2×A4
LCSG ▷ Q8
UCSC1 ◁ C2×C4

49 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 2 generators on 18 points:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11711057, #11711058, #11711059, …
maximal subgroup ofC2×GU(2,𝔽3), #5423, #5424, …
maximal normal subgroup ofC2×GU(2,𝔽3), #5423, #5424, …
maximal quotient ofC2×C4×SL(2,𝔽3), C4×#41⋊C3, #548⋊C3, …
outer automorphism group of#1308

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 200);