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C5×C3⋊(ΓL(1,𝔽8)⋊Dic17) #36339306

order21420
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×C3⋊(ΓL(1,𝔽8)⋊Dic17)
unambiguous semidirect productC3⋊(C5×ΓL(1,𝔽8)⋊Dic17)
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×Dic51)
unambiguous semidirect productDic357⋊(C3×C5)
unambiguous semidirect product(C5×Dic357)⋊C3

Splittings

derivedC3×C7×C17 ⋊ C3×C4×C5
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C4
HallC3×C17×ΓL(1,𝔽8) ⋊ C4×C5
HallC5×C7×C17 ⋊ C3×Dic3
HallC3×C5×ΓL(1,𝔽8) ⋊ Dic17
HallC7×C17 ⋊ C3×C5×Dic3
HallC5×C17 ⋊ C3⋊(ΓL(1,𝔽8)⋊C4)
HallC3×ΓL(1,𝔽8) ⋊ C5×Dic17
HallC5×C7 ⋊ C3×Dic51
SylowC17 ⋊ C5×C3⋊(ΓL(1,𝔽8)⋊C4)
SylowC7 ⋊ C3×C5×Dic51

Subgroups

subgroupG/H
center, hypercenterC2×C5C3⋊(C17⋊AGL(1,𝔽7))
commutator, hypocenterC3×C7×C17C3×C4×C5
FrattiniC2C5×C3⋊(C17⋊AGL(1,𝔽7))
Fitting, socleC2×C3×C5×C7×C17C2×C3
prime radicalC2×C3×C5×C17×ΓL(1,𝔽8)C2
cosocleC2×C3×C7×C17C2×C3×C5
LCSG ▷ C3×C7×C17
UCSC1 ◁ C2×C5

112 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 7140

Conjugacy classes & character table

1320 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 138);