Finite Group Databasebeta

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Cayley diagram

C8×C22⋊C9 #63428

order288
isoclinismisoclinic class of A4
automorphismAut(G) = C22×C3×S4 (order 288) · Inn(G) = A4 · Out(G) = C23×C3
homologyAbelianization (H1) = C8×C9 · Schur multiplier (H2) = C2

Aliases

direct productC8×C22⋊C9
unambiguous semidirect product(C22×C8)⋊C9

Splittings

derivedC22 ⋊ C8×C9
SylowC22×C8 ⋊ C9

Subgroups

subgroupG/H
center, hypercenterC3×C8A4
commutator, hypocenterC22C8×C9
FrattiniC3×C4C2×A4
FittingC22×C3×C8C3
socle, prime radicalC23×C3C3×C4
cosocleC22×C3×C4C2×C3
LCSG ▷ C22
UCSC1 ◁ C3×C8

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 2 generators on 21 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11919988, #11919991, #11919992, …
maximal subgroup of(C22⋊C9)⋊C16, C16×C22⋊C9, #10589500, …
maximal normal subgroup of(C22⋊C9)⋊C16, C16×C22⋊C9, #10589500, …
maximal quotient ofC8×Q8⋊C9, C16×C22⋊C9, #504⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 76);