Finite Group Databasebeta

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Cayley diagram

C4×#942 #82427

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

Aliases

direct productC4×#942

Splittings

FittingC2×C3×C4×Q8 ⋊ C2
SylowC3 ⋊ C4×#137

Subgroups

subgroupG/H
centerC2×C4C2×#67
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC2×C3×C4×Q8C2
socleC22×C3C22×D4
prime radicalC2×#235C22
cosocleC2×C3×C4C24
hypercenterC2×C4×Q8D3
hypocenterC3C4×#137
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×Q8

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10689801, #10689807, #10689815, …
maximal normal subgroup of#10689801, #10689807, #10689815, …
maximal quotient ofC8×#942, #10764781, #10764801, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 15003);