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Cayley diagram

D4×SL(2,𝔽3) #5445

order192
isoclinismisoclinic class of #548⋊C3
automorphismAut(G) = S4×#120 (order 768) · Inn(G) = C22×A4 · Out(G) = C2×D4
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productD4×SL(2,𝔽3)
unambiguous semidirect productC4⋊(C2×SL(2,𝔽3))
unambiguous semidirect product(D4×Q8)⋊C3

Splittings

Fitting, SylowD4×Q8 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, socleC22C22×A4
commutator, cosocleC2×Q8C22×C3
FittingD4×Q8C3
hypercenterC2×D4A4
hypocenterQ8C3×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ C2×D4

105 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:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#73085, #73086, #73087, …
maximal normal subgroup of#73085, #73086, #73087, …
maximal quotient ofC22⋊C4×SL(2,𝔽3), C4⋊C4×SL(2,𝔽3), D8×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1004);