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Cayley diagram

C2×Q8×SL(2,𝔽3) #85599

order384
isoclinismisoclinic class of #548⋊C3
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC2×Q8×SL(2,𝔽3)

Splittings

Fitting, SylowC2×Q82 ⋊ C3

Subgroups

subgroupG/H
center, socleC23C22×A4
commutator, cosocleC2×Q8C23×C3
FrattiniC22C23×A4
FittingC2×Q82C3
prime radicalC22×SL(2,𝔽3)C22
hypercenterC22×Q8A4
hypocenterQ8C2×C3×Q8
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C23 ◁ C22×Q8

347 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofSL(2,𝔽3)×#101, SL(2,𝔽3)×#103, C4×Q8×SL(2,𝔽3), …
maximal normal subgroup ofSL(2,𝔽3)×#101, SL(2,𝔽3)×#103, C4×Q8×SL(2,𝔽3), …
maximal quotient ofC2×C4⋊C4×SL(2,𝔽3), C4×Q8×SL(2,𝔽3), SL(2,𝔽3)×C22⋊Q8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18175);