Finite Group Databasebeta

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Cayley diagram

C4×#898 #73621

order384
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C24

Aliases

direct productC4×#898

Splittings

FittingC3×C4×C4⋊C4 ⋊ C2
SylowC3 ⋊ C4×#126

Subgroups

subgroupG/H
centerC22×C4C22×D3
commutatorC22×C3C23×C4
FrattiniC23C23×D3
FittingC3×C4×C4⋊C4C2
socle, cosocleC23×C3C24
prime radicalC23×D3C23
hypercenterC4×C4⋊C4D3
hypocenterC3C4×#126
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C4

516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

(1,2,3)(9,13)(12,15)(16,18,19,22)(17,21,20,23);(8,10,11,14)(9,12,13,15)(17,20)(18,22);(2,3)(8,9)(10,12)(11,13)(14,15)(16,17)(18,21)(19,20)(22,23);(4,5,6,7)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10803399, C8×#898, #10803406, …
maximal normal subgroup of#10803399, C8×#898, #10803406, …
maximal quotient ofC4×#4670, C4×#4669, C4×#4990, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 6197);