Finite Group Databasebeta

← Prev (#63168) Next (#63170) →

Cayley diagram

C22×D3×C11 #63169

order264
isoclinismisoclinic class of D3
automorphismAut(G) = C2×C5×D3×S4 (order 1440) · Inn(G) = D3 · Out(G) = C2×C5×S4
homologyAbelianization (H1) = C23×C11 · Schur multiplier (H2) = C23

Aliases

direct productC22×D3×C11
unambiguous semidirect productC3⋊(C23×C11)
unambiguous semidirect product(C2×C3)⋊(C22×C11)

Splittings

Fitting, socleC22×C3×C11 ⋊ C2
derived, SylowC3 ⋊ C23×C11
HallC3×C11 ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC22×C11D3
commutator, cosocle, hypocenterC3C23×C11
FrattiniC1G
Fitting, socleC22×C3×C11C2
LCSG ▷ C3
UCSC1 ◁ C22×C11

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 132

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×D3)⋊Dic11, C11×(C2×D3)⋊C4, C2×D3×Dic11, …
maximal normal subgroup of(C2×D3)⋊Dic11, C11×(C2×D3)⋊C4, C2×D3×Dic11, …
maximal quotient ofC2×C11×Dic6, C2×C4×D3×C11, C2×C11×D12, …
automorphism group ofDic3×C23, C2×D3×C23, C22×C3×C23, …
outer automorphism group ofC22×C3×C23, #41×C23

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(264, 37);