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Cayley diagram

C2×Q8⋊SL(2,𝔽3) #85652

order384
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C22

Aliases

direct productC2×Q8⋊SL(2,𝔽3)

Splittings

Fitting, SylowC2×Q82 ⋊ C3
derivedQ82 ⋊ C2×C3

Subgroups

subgroupG/H
center, socle, hypercenterC23C22⋊A4
commutator, cosocle, hypocenterQ82C2×C3
FrattiniC22C2×C22⋊A4
FittingC2×Q82C3
LCSG ▷ Q82
UCSC1 ◁ C23

257 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)(12,13,15)(16,18,17);(3,7,6,9)(4,10,8,5);(11,15,14,17)(12,18,16,13)

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72228026, #72261377
maximal subgroup of#12777⋊C3, C2×#2205⋊C3, C4×Q8⋊SL(2,𝔽3), …
maximal normal subgroup of#12777⋊C3, C2×#2205⋊C3, C4×Q8⋊SL(2,𝔽3), …
maximal quotient ofC4×Q8⋊SL(2,𝔽3), C2×#73291, #52287⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18228);