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AGL(3,𝔽2) #13062397

order1344
isoclinismisoclinic class of AGL(3,𝔽2)
automorphismAut(G) = #72227515 (order 2688) · Inn(G) = G · Out(G) = C2
homologyAbelianization (H1) = C1 · Schur multiplier (H2) = C22

Aliases

affine general linearAGL(3,𝔽2)
unambiguous semidirect productC23⋊GL(3,𝔽2)

Splittings

Fitting, socleC23 ⋊ GL(3,𝔽2)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, hypocenterGC1
Fitting, socle, cosocleC23GL(3,𝔽2)
LCSG
UCSC1

95 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 8
transitive degreeμt(G) = 8

A minimal faithful permutation representation — 2 generators on 8 points:

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

Conjugacy classes & character table

C1C2C3C4C5C6C7C8C9C10C11
size17424284168168192192224224
order12224447736
χ111111111111
χ233-1-1-111−1+i72−1−i7200
χ333-1-1-111−1−i72−1+i7200
χ46622200-1-100
χ577-1-1-1-1-10011
χ67-13-1-11-1001-1
χ77-1-13-1-11001-1
χ8880000011-1-1
χ914-222-20000-11
χ1021-31-31-110000
χ1121-3-3111-10000

Here i is the imaginary unit (i² = −1).

Relationships

derived subgroup of#72227515, C3×C4×AGL(3,𝔽2), C2×C5×AGL(3,𝔽2), …
maximal subgroup of#72209643, #72209649, #72209654, …
maximal normal subgroup ofC2×AGL(3,𝔽2), C3×AGL(3,𝔽2), C5×AGL(3,𝔽2), …
maximal quotient of#72209595, #72209596, #72209643, …
outer automorphism group ofC23×D3, C23×AGL(1,𝔽5), C23×S4
holomorph ofC23

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11686);