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C3×C5×C17×ΓL(1,𝔽8)⋊C4 #36339298

order21420
automorphism?
homologyAbelianization (H1) = C32×C4×C5×C17 · Schur multiplier (H2) = C3

Aliases

direct productC3×C5×C17×ΓL(1,𝔽8)⋊C4
unambiguous semidirect productΓL(1,𝔽8)⋊(C3×C4×C5×C17)
unambiguous semidirect productDic7⋊(C32×C5×C17)
unambiguous semidirect product(C3×Dic7)⋊(C3×C5×C17)
unambiguous semidirect product(C5×Dic7)⋊(C32×C17)
unambiguous semidirect product(C3×C5×Dic7)⋊(C3×C17)
unambiguous semidirect product(C17×Dic7)⋊(C32×C5)
unambiguous semidirect product(C3×C17×Dic7)⋊(C3×C5)
unambiguous semidirect product(C5×C17×Dic7)⋊C32
unambiguous semidirect product(C3×C5×C17×Dic7)⋊C3

Splittings

derived, SylowC7 ⋊ C32×C4×C5×C17
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C4
HallC5×C17×Dic7 ⋊ C32
HallC3×C17×ΓL(1,𝔽8) ⋊ C4×C5
HallC5×C7×C17 ⋊ C32×C4
HallC17×Dic7 ⋊ C32×C5
HallC3×C5×ΓL(1,𝔽8) ⋊ C4×C17
HallC5×Dic7 ⋊ C32×C17
HallC7×C17 ⋊ C32×C4×C5
HallC3×ΓL(1,𝔽8) ⋊ C4×C5×C17
HallC5×C7 ⋊ C32×C4×C17
HallDic7 ⋊ C32×C5×C17

Subgroups

subgroupG/H
center, hypercenterC2×C3×C5×C17AGL(1,𝔽7)
commutator, hypocenterC7C32×C4×C5×C17
FrattiniC2C3×C5×C17×AGL(1,𝔽7)
Fitting, socleC2×C3×C5×C7×C17C2×C3
prime radicalC2×C3×C5×C17×ΓL(1,𝔽8)C2
cosocleC2×C7C2×C32×C5×C17
LCSG ▷ C7
UCSC1 ◁ C2×C3×C5×C17

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 7140

Conjugacy classes & character table

3570 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 130);