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C5×C17×C3⋊(ΓL(1,𝔽8)⋊C4) #36339305

order21420
automorphism?
homologyAbelianization (H1) = C3×C4×C5×C17 · Schur multiplier (H2) = C1

Aliases

direct productC5×C17×C3⋊(ΓL(1,𝔽8)⋊C4)
unambiguous semidirect productC3⋊(C5×C17×ΓL(1,𝔽8)⋊C4)
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×Dic3×C17)
unambiguous semidirect productDic21⋊(C3×C5×C17)
unambiguous semidirect product(C5×Dic21)⋊(C3×C17)
unambiguous semidirect product(C17×Dic21)⋊(C3×C5)
unambiguous semidirect product(C5×C17×Dic21)⋊C3

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC2×C5×C17C3⋊AGL(1,𝔽7)
commutator, hypocenterC3×C7C3×C4×C5×C17
FrattiniC2C5×C17×C3⋊AGL(1,𝔽7)
Fitting, socleC2×C3×C5×C7×C17C2×C3
prime radicalC2×C3×C5×C17×ΓL(1,𝔽8)C2
cosocleC2×C3×C7C2×C3×C5×C17
LCSG ▷ C3×C7
UCSC1 ◁ C2×C5×C17

112 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 56 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

2040 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 137);