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Cayley diagram

C22×#41⋊C3 #5941

order192
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = S4×#5934 (order 4608) · Inn(G) = A4 · Out(G) = C2×#5934
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC22×#41⋊C3

Splittings

Fitting, SylowC22×#41 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC22×C4A4
commutator, cosocle, hypocenterQ8C23×C3
FrattiniC2C23×A4
FittingC22×#41C3
socleC23C2×A4
LCSG ▷ Q8
UCSC1 ◁ C22×C4

220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2)(3,4)(5,6,9,12)(7,11,15,18,8,17,14,20,13,10,16,19);(3,4)(5,13,9,15)(6,17,12,19)(7,16,14,8)(10,20,18,11);(1,2)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72268142, #72268160, #72268912, …
maximal subgroup of#73149, #73151, #2341⋊C3, …
maximal normal subgroup of#73149, #73151, #2341⋊C3, …
maximal quotient ofC22×C4×SL(2,𝔽3), C2×C4×#41⋊C3, C2×#548⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1500);