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Cayley diagram

Q8×AGL(1,ℤ/9) #88723

order432
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productQ8×AGL(1,ℤ/9)
unambiguous semidirect productD9⋊(C3×Q8)

Splittings

FittingQ8×3−1+2 ⋊ C2
Sylow3−1+2 ⋊ C2×Q8

Subgroups

subgroupG/H
centerC2C22×AGL(1,ℤ/9)
commutator, cosocleC2×C9C23×C3
Frattini, socleC2×C3C22×C3×D3
FittingQ8×3−1+2C2
prime radicalC2×AGL(1,ℤ/9)C22
hypercenterQ8AGL(1,ℤ/9)
hypocenterC9C2×C3×Q8
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ Q8

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 72

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

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofSD16×AGL(1,ℤ/9), #63477⋊C3, Q16×AGL(1,ℤ/9), …
maximal normal subgroup ofSD16×AGL(1,ℤ/9), #63477⋊C3, Q16×AGL(1,ℤ/9), …
maximal quotient of#63449⋊C3, #63450⋊C3, #63451⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 370);