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Cayley diagram

C2×3−1+2⋊C8 #88495

order432
isoclinismisoclinic class of AGL(1,ℤ/9)
automorphismAut(G) = C2×D4×AGL(1,ℤ/9) (order 864) · Inn(G) = AGL(1,ℤ/9) · Out(G) = C2×D4
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2

Aliases

direct productC2×3−1+2⋊C8
unambiguous semidirect product(C2×3−1+2)⋊C8
unambiguous semidirect product(C2×C9⋊C8)⋊C3

Splittings

derivedC9 ⋊ C2×C3×C8
Sylow3−1+2 ⋊ C2×C8

Subgroups

subgroupG/H
center, hypercenterC2×C4AGL(1,ℤ/9)
commutator, hypocenterC9C2×C3×C8
FrattiniC3×C4C2×C3×D3
FittingC2×C4×3−1+2C2
socleC22×C3C3×Dic3
prime radicalC22×C32Dic3
cosocleC4×C9C22×C3
LCSG ▷ C9
UCSC1 ◁ C2×C4

74 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:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×3−1+2⋊C8, #63362⋊C3, C4⋊(3−1+2⋊C8), …
maximal normal subgroup ofC4×3−1+2⋊C8, #63362⋊C3, C4⋊(3−1+2⋊C8), …
maximal quotient ofC4×3−1+2⋊C8, C4⋊(3−1+2⋊C8), C2×3−1+2⋊C16, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 142);