Finite Group Databasebeta

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Cayley diagram

C2×#5494 #85695

order384
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#5494

Splittings

FittingC22×C3×C22⋊C4 ⋊ C2
SylowC3 ⋊ C2×#538

Subgroups

subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC22×C3×C22⋊C4C2
socleC23×C3C24
hypercenterC22×C22⋊C4D3
hypocenterC3C2×#538
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C22⋊C4

1336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 5 generators on 21 points:

(1,2)(3,4,5)(6,8)(7,10)(9,12)(11,13)(14,16)(15,19)(17,20)(18,21);(6,7,9,11)(8,10,12,13)(16,20)(19,21);(14,15)(16,19)(17,18)(20,21);(4,5)(8,12)(10,13)(15,18)(19,21);(1,2)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10687015, #10687019, #10761207, …
maximal normal subgroup of#10687015, #10687019, #10761207, …
maximal quotient ofC2×#82093, C2×#73367, C2×#82096, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18271);