Finite Group Databasebeta

← Prev (#88683) Next (#88685) →

Cayley diagram

C3×C4×C22⋊C9 #88684

order432
isoclinismisoclinic class of A4
automorphism?
homologyAbelianization (H1) = C3×C4×C9 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3×C4×C22⋊C9
unambiguous semidirect product(C22×C3×C4)⋊C9

Splittings

derivedC22 ⋊ C3×C4×C9
SylowC22×C4 ⋊ C3×C9

Subgroups

subgroupG/H
center, hypercenterC32×C4A4
commutator, hypocenterC22C3×C4×C9
FrattiniC2×C3C2×C3×A4
FittingC22×C32×C4C3
socle, prime radicalC23×C32C2×C3
cosocleC23×C3C2×C32
LCSG ▷ C22
UCSC1 ◁ C32×C4

102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13146263, #13146268, #13146269, …
maximal subgroup ofC3×(C22⋊C9)⋊C8, C3⋊((C22⋊C9)⋊C8), C3⋊C8×C22⋊C9, …
maximal normal subgroup ofC3×(C22⋊C9)⋊C8, C3⋊((C22⋊C9)⋊C8), C3⋊C8×C22⋊C9, …
maximal quotient ofC3×C4×Q8⋊C9, C3×C8×C22⋊C9, C3×#131⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 331);