Finite Group Databasebeta

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Cayley diagram

C22×3−1+2⋊C4 #88731

order432
isoclinismisoclinic class of AGL(1,ℤ/9)
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC22×3−1+2⋊C4

Splittings

derivedC9 ⋊ C22×C3×C4
Sylow3−1+2 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC23AGL(1,ℤ/9)
commutator, hypocenterC9C22×C3×C4
FrattiniC2×C3C22×C3×D3
FittingC23×3−1+2C2
socleC23×C3C3×D3
prime radicalC23×C32D3
cosocleC2×C9C23×C3
LCSG ▷ C9
UCSC1 ◁ C23

178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of3−1+2⋊#95, #63439⋊C3, #63440⋊C3, …
maximal normal subgroup of3−1+2⋊#95, #63439⋊C3, #63440⋊C3, …
maximal quotient ofC22×3−1+2⋊C8, C2×#3667⋊C3, C2×C4×3−1+2⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 378);