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Cayley diagram

D4×#157 #64288

order288
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C3

Aliases

direct productD4×#157

Splittings

FittingC32×D4 ⋊ C4
SylowC32 ⋊ C4×D4

Subgroups

subgroupG/H
center, FrattiniC2C22×#157
commutator, socleC2×C32C22×C4
FittingC32×D4C4
prime radicalD4×C3⋊D3C2
cosocleC2×C3⋊D3C23
hypercenterD4#157
hypocenterC32C4×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ D4

148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 3 generators on 10 points:

(1,2)(3,5,6,4);(1,3,6)(2,5,4)(8,10);(7,8)(9,10)

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD4⋊AGL(1,𝔽9), #10593462, #10593464, …
maximal normal subgroup ofD4⋊AGL(1,𝔽9), #10593462, #10593464, …
maximal quotient of#10593739, #10593740, #10593741, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 936);