Finite Group Databasebeta

← Prev (#79584) Next (#79586) →

Cayley diagram

C4×#912 #79585

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

Aliases

direct productC4×#912

Splittings

FittingC3×C4×D8 ⋊ C2
SylowC3 ⋊ C4×#135

Subgroups

subgroupG/H
centerC2×C4D3×D4
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC3×C4×D8C2
socleC22×C3C22×D4
prime radicalC2×#912C2
cosocleC2×C3×C4C24
hypercenterC4×D8D3
hypocenterC3C4×#135
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×D8

500 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)(10,17)(11,20)(13,21)(16,23)(18,19);(8,10)(9,13)(11,17)(12,19)(14,16)(15,23)(18,20)(21,22);(2,3)(8,9)(10,16)(11,15)(12,14)(13,19)(17,21)(18,23)(20,22);(4,5,6,7)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10704924, #10704929, #10704950, …
maximal normal subgroup of#10704924, #10704929, #10704950, …
maximal quotient ofC8×#912, #10764711, #10764931, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12161);