Finite Group Databasebeta

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Cayley diagram

C4×#915 #79646

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

Aliases

direct productC4×#915

Splittings

FittingC3×C4×SD16 ⋊ C2
SylowC3 ⋊ C4×#137

Subgroups

subgroupG/H
centerC2×C4D3×D4
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC3×C4×SD16C2
socleC22×C3C22×D4
prime radicalC2×#237C22
cosocleC2×C3×C4C24
hypercenterC4×SD16D3
hypocenterC3C4×#137
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×SD16

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×#915, #10764615, #10764629, …
maximal normal subgroup ofC8×#915, #10764615, #10764629, …
maximal quotient ofC8×#915, #10764776, #10764798, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12222);