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Cayley diagram

Q8×SL(2,𝔽3) #5448

order192
isoclinismisoclinic class of #548⋊C3
automorphismAut(G) = S4×#1020 (order 2304) · Inn(G) = C22×A4 · Out(G) = C2×S4
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productQ8×SL(2,𝔽3)

Splittings

Fitting, SylowQ82 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, socleC22C22×A4
commutator, cosocleC2×Q8C22×C3
FittingQ82C3
prime radicalC2×SL(2,𝔽3)C22
hypercenterC2×Q8A4
hypocenterQ8C3×Q8
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22 ◁ C2×Q8

77 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

derived subgroup ofSL(2,𝔽3)×#226, #85430⋊C3, #85428⋊C3, …
maximal subgroup of#73097, #73098, #73099, …
maximal normal subgroup of#73097, #73098, #73099, …
maximal quotient ofC4⋊C4×SL(2,𝔽3), C2×Q8×SL(2,𝔽3), Q8×Q8⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1007);