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B(4,𝔽2)×AGL(2,𝔽3) #72253281

order27648
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

direct productB(4,𝔽2)×AGL(2,𝔽3)

Splittings

FittingC32×B(4,𝔽2) ⋊ GL(2,𝔽3)

Subgroups

subgroupG/H
centerC2#120×AGL(2,𝔽3)
commutator, cosocleC23×PSU(3,𝔽2)⋊C3C24
FrattiniC23C23×AGL(2,𝔽3)
FittingC32×B(4,𝔽2)GL(2,𝔽3)
socleC2×C32#120×GL(2,𝔽3)
hypercenterB(4,𝔽2)AGL(2,𝔽3)
hypocenterPSU(3,𝔽2)⋊C3C2×B(4,𝔽2)
LCSG ▷ C23×PSU(3,𝔽2)⋊C3 ▷ C2×PSU(3,𝔽2)⋊C3 ▷ PSU(3,𝔽2)⋊C3
UCSC1 ◁ C2 ◁ C23 ◁ B(4,𝔽2)

At least 7926 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

automorphism group ofC2×#634, C2×C4⋊(C3⋊D3)
holomorph ofC2×C32×C4

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,9,4,5,3,7,6), (2,5,4)(7,8,9)(14,17), (10,12,13,16)(11,14,15,17), (10,11)(12,14)(13,15)(16,17));