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Cayley diagram

C23×SL(2,𝔽3) #5939

order192
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) order 32256
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC23×SL(2,𝔽3)

Splittings

Fitting, SylowC23×Q8 ⋊ C3
derivedQ8 ⋊ C23×C3

Subgroups

subgroupG/H
center, socle, hypercenterC24A4
commutator, cosocle, hypocenterQ8C23×C3
FrattiniC2C23×A4
FittingC23×Q8C3
LCSG ▷ Q8
UCSC1 ◁ C24

280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72241493, #72241524, #72241527, …
maximal subgroup of#73148, C22⋊C4×SL(2,𝔽3), C22×#859, …
maximal normal subgroup of#73148, C22⋊C4×SL(2,𝔽3), C22×#859, …
maximal quotient ofC22×C4×SL(2,𝔽3), C2×D4×SL(2,𝔽3), C2×Q8×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1498);