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C3×C5×ΓL(1,𝔽8)⋊Dic17 #36339299

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

Aliases

direct productC3×C5×ΓL(1,𝔽8)⋊Dic17
unambiguous semidirect productΓL(1,𝔽8)⋊(C3×C5×Dic17)
unambiguous semidirect productDic119⋊(C32×C5)
unambiguous semidirect product(C3×Dic119)⋊(C3×C5)
unambiguous semidirect product(C5×Dic119)⋊C32
unambiguous semidirect product(C3×C5×Dic119)⋊C3

Splittings

derived, HallC7×C17 ⋊ C32×C4×C5
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C4
HallC5×Dic119 ⋊ C32
HallC3×C17×ΓL(1,𝔽8) ⋊ C4×C5
HallC5×C7×C17 ⋊ C32×C4
HallDic119 ⋊ C32×C5
HallC3×C5×ΓL(1,𝔽8) ⋊ Dic17
HallC5×C17 ⋊ C3×ΓL(1,𝔽8)⋊C4
HallC3×ΓL(1,𝔽8) ⋊ C5×Dic17
HallC5×C7 ⋊ C32×Dic17
SylowC17 ⋊ C3×C5×ΓL(1,𝔽8)⋊C4
SylowC7 ⋊ C32×C5×Dic17

Subgroups

subgroupG/H
center, hypercenterC2×C3×C5C17⋊AGL(1,𝔽7)
commutator, hypocenterC7×C17C32×C4×C5
FrattiniC2C3×C5×C17⋊AGL(1,𝔽7)
Fitting, socleC2×C3×C5×C7×C17C2×C3
prime radicalC2×C3×C5×C17×ΓL(1,𝔽8)C2
cosocleC2×C7×C17C2×C32×C5
LCSG ▷ C7×C17
UCSC1 ◁ C2×C3×C5

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 7140

Conjugacy classes & character table

1410 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 131);