Finite Group Databasebeta

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Cayley diagram

C4×#909 #78536

order384
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC4×#909

Splittings

FittingC3×C4×M16 ⋊ C2
SylowC3 ⋊ C4×#137

Subgroups

subgroupG/H
centerC2×C4C2×D12
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC3×C4×M16C2
socleC22×C3C22×D4
prime radicalC2×#235C22
cosocleC2×C3×C4C24
hypercenterC4×M16D3
hypocenterC3C4×#137
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×M16

484 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)(8,11,12,19,13,20,10,17)(9,15,14,18,16,23,21,22);(8,10,13,12)(9,14,16,21)(11,19,20,17)(15,22,23,18);(2,3)(8,9)(10,14)(11,18)(12,21)(13,16)(15,19)(17,23)(20,22);(4,5,6,7)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10689803, #10689805, #10689817, …
maximal normal subgroup of#10689803, #10689805, #10689817, …
maximal quotient ofC8×#909, #10764785, #10764807, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11112);