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Cayley diagram

C11×C22⋊C9 #87674

order396
isoclinismisoclinic class of A4
automorphismAut(G) = C2×C3×C5×S4 (order 720) · Inn(G) = A4 · Out(G) = C22×C3×C5
homologyAbelianization (H1) = C9×C11 · Schur multiplier (H2) = C2

Aliases

direct productC11×C22⋊C9
unambiguous semidirect productC22⋊(C9×C11)
unambiguous semidirect product(C22×C11)⋊C9

Splittings

derived, SylowC22 ⋊ C9×C11
HallC22×C11 ⋊ C9

Subgroups

subgroupG/H
center, hypercenterC3×C11A4
commutator, hypocenterC22C9×C11
FrattiniC3C11×A4
Fitting, socle, prime radicalC22×C3×C11C3
cosocleC22×C3C3×C11
LCSG ▷ C22
UCSC1 ◁ C3×C11

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 198

A minimal faithful permutation representation — 2 generators on 24 points:

(2,3,4)(5,9,13,7,12,8,10,6,11)(14,15,16,17,18,19,20,21,22,23,24);(1,4)(2,3)

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC11⋊((C22⋊C9)⋊C2), C11⋊((C22⋊C9)⋊C4), D11×(C22⋊C9)⋊C2, …
maximal subgroup ofC11×(C22⋊C9)⋊C2, C11⋊((C22⋊C9)⋊C2), D11×C22⋊C9, …
maximal normal subgroup ofC11×(C22⋊C9)⋊C2, C11⋊((C22⋊C9)⋊C2), D11×C22⋊C9, …
maximal quotient ofC11×Q8⋊C9, C2×C11×C22⋊C9, C11×C22⋊C27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(396, 6);