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Cayley diagram

C2×#5464 #85655

order384
isoclinismisoclinic class of #5464
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C42

Aliases

direct productC2×#5464

Splittings

Fitting, SylowC2×H(3,𝔽4) ⋊ C3
derivedH(3,𝔽4) ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2#5464
commutator, cosocle, hypocenterH(3,𝔽4)C2×C3
FrattiniC22C2×C22⋊A4
FittingC2×H(3,𝔽4)C3
socleC23C22⋊A4
LCSG ▷ H(3,𝔽4)
UCSC1 ◁ C2

288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 24

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

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

Conjugacy classes & character table

18 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72261513, #72261518, #72261576, …
maximal subgroup of#8506⋊C3, C2×#1655⋊C3, C4×#5464, …
maximal normal subgroup of#8506⋊C3, C2×#1655⋊C3, C4×#5464, …
maximal quotient ofC4×#5464, C2×#73292, #11710040, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18231);