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Cayley diagram

C4×SL(2,𝔽3) #862

order96
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C22×S4 (order 96) · Inn(G) = A4 · Out(G) = C23
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC4×SL(2,𝔽3)
unambiguous semidirect product(C4×Q8)⋊C3

Splittings

Fitting, SylowC4×Q8 ⋊ C3
derivedQ8 ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC2×C4A4
commutator, hypocenterQ8C3×C4
Frattini, socleC22C2×A4
FittingC4×Q8C3
prime radicalC2×SL(2,𝔽3)C2
cosocleC2×Q8C2×C3
LCSG ▷ Q8
UCSC1 ◁ C2×C4

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

derived subgroup of#73011, #73012, #73013, …
maximal subgroup ofSL(2,𝔽3)⋊C8, C8×SL(2,𝔽3), #5386, …
maximal normal subgroup ofSL(2,𝔽3)⋊C8, C8×SL(2,𝔽3), #5386, …
maximal quotient ofC8×SL(2,𝔽3), C2×C4×SL(2,𝔽3), C4×Q8⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 69);