Finite Group Databasebeta

← Prev (#88687) Next (#88689) →

Cayley diagram

C2×#6204 #88688

order432
isoclinismisoclinic class of C9⋊SL(2,𝔽3)
automorphismAut(G) = C2×A4⋊#272 (order 1296) · Inn(G) = C3×A4 · Out(G) = C2×C3⋊D3
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C1

Aliases

direct productC2×#6204

Splittings

SylowC2×Q8 ⋊ 3−1+2

Subgroups

subgroupG/H
center, socleC22×C3C3×A4
commutator, cosocleC3×Q8C2×C32
FrattiniC2×C3C2×C3×A4
FittingC2×C32×Q8C3
prime radical, hypercenterC22×C32A4
hypocenterQ8C2×3−1+2
LCSG ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C22×C32

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#63421⋊C3, C4×#6204, C2×#3688⋊C3, …
maximal normal subgroup of#63421⋊C3, C4×#6204, C2×#3688⋊C3, …
maximal quotient ofC4×#6204, C22×#6204, C2×(Q8⋊C9)⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 335);