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Cayley diagram

C22×GO+(4,𝔽2) #64383

order288
isoclinismisoclinic class of GO+(4,𝔽2)
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC22×GO+(4,𝔽2)
unambiguous semidirect product#157⋊C23
unambiguous semidirect productD32⋊C23

Splittings

Fitting, socleC22×C32 ⋊ D4
SylowC32 ⋊ C22×D4

Subgroups

subgroupG/H
center, hypercenterC22GO+(4,𝔽2)
commutator, cosocleC3⋊D3C24
FrattiniC1G
Fitting, socleC22×C32D4
hypocenterC32C22×D4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C22

370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 10 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10593453, #10593454, #10593455, …
maximal normal subgroup of#10593453, #10593454, #10593455, …
maximal quotient ofC2×D32⋊Q8, C2×#64221, #10596494, …
automorphism group ofD32⋊C4, #3775

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1031);