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B(4,𝔽2)×#64425 #72246206

order18816
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C27×C7

Aliases

direct productB(4,𝔽2)×#64425

Splittings

FittingC72×B(4,𝔽2) ⋊ C2×C3
HallB(4,𝔽2)×C7⋊D7 ⋊ C3
Hall#3862 ⋊ C2×B(4,𝔽2)
SylowC72 ⋊ C2×C3×B(4,𝔽2)

Subgroups

subgroupG/H
centerC2#120×#64425
commutator, cosocleC23×C72C24×C3
FrattiniC23C23×#64425
FittingC72×B(4,𝔽2)C2×C3
socleC2×C72C2×C3×#120
hypercenterB(4,𝔽2)#64425
hypocenterC72C2×C3×B(4,𝔽2)
LCSG ▷ C23×C72 ▷ C2×C72 ▷ C72
UCSC1 ◁ C2 ◁ C23 ◁ B(4,𝔽2)

5090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofB(4,𝔽2)×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
maximal normal subgroup ofB(4,𝔽2)×ΓL(1,𝔽8)⋊AGL(1,𝔽7)

GAP

Past the SmallGroups library coverage — no SmallGroups id.