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Cayley diagram

C4×GO+(4,𝔽2) #64229

order288
isoclinismisoclinic class of GO+(4,𝔽2)
automorphismAut(G) = C2×#10596417 (order 1152) · Inn(G) = GO+(4,𝔽2) · Out(G) = C2×D4
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC4×GO+(4,𝔽2)

Splittings

FittingC32×C4 ⋊ D4
SylowC32 ⋊ C4×D4

Subgroups

subgroupG/H
center, hypercenterC4GO+(4,𝔽2)
commutatorC3⋊D3C22×C4
FrattiniC2C2×GO+(4,𝔽2)
FittingC32×C4D4
socleC2×C32C2×D4
prime radicalC2×GO+(4,𝔽2)C2
cosocleC2×C3⋊D3C23
hypocenterC32C4×D4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C4

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:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofGO+(4,𝔽2)⋊C8, C8×GO+(4,𝔽2), #10593282, …
maximal normal subgroup ofGO+(4,𝔽2)⋊C8, C8×GO+(4,𝔽2), #10593282, …
maximal quotient ofC8×GO+(4,𝔽2), #10593282, #10593283, …
automorphism group ofC5×D32
outer automorphism group ofC5×H(5,𝔽2)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 877);