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C3×ΓL(1,𝔽8)⋊Dic85 #36339302

order21420
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productC3×ΓL(1,𝔽8)⋊Dic85
unambiguous semidirect productΓL(1,𝔽8)⋊(C3×Dic85)
unambiguous semidirect productDic595⋊C32
unambiguous semidirect product(C3×Dic595)⋊C3

Splittings

derived, HallC5×C7×C17 ⋊ C32×C4
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C4
HallDic595 ⋊ C32
HallC3×C17×ΓL(1,𝔽8) ⋊ Dic5
HallC3×C5×ΓL(1,𝔽8) ⋊ Dic17
HallC7×C17 ⋊ C32×Dic5
HallC5×C17 ⋊ C3×ΓL(1,𝔽8)⋊C4
HallC3×ΓL(1,𝔽8) ⋊ Dic85
HallC5×C7 ⋊ C32×Dic17
SylowC17 ⋊ C3×ΓL(1,𝔽8)⋊Dic5
SylowC7 ⋊ C32×Dic85
SylowC5 ⋊ C3×ΓL(1,𝔽8)⋊Dic17

Subgroups

subgroupG/H
center, hypercenterC2×C3C5⋊(C17⋊AGL(1,𝔽7))
commutator, hypocenterC5×C7×C17C32×C4
FrattiniC2C3×C5⋊(C17⋊AGL(1,𝔽7))
Fitting, socleC2×C3×C5×C7×C17C2×C3
prime radicalC2×C3×C5×C17×ΓL(1,𝔽8)C2
cosocleC2×C5×C7×C17C2×C32
LCSG ▷ C5×C7×C17
UCSC1 ◁ C2×C3

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 7140

Conjugacy classes & character table

1302 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 134);