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C17×C3⋊(ΓL(1,𝔽8)⋊Dic5) #36339309

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

Aliases

direct productC17×C3⋊(ΓL(1,𝔽8)⋊Dic5)
unambiguous semidirect productC3⋊(C17×ΓL(1,𝔽8)⋊Dic5)
unambiguous semidirect productΓL(1,𝔽8)⋊(C17×Dic15)
unambiguous semidirect productDic105⋊(C3×C17)
unambiguous semidirect product(C17×Dic105)⋊C3

Splittings

derivedC3×C5×C7 ⋊ C3×C4×C17
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C4
HallC3×C17×ΓL(1,𝔽8) ⋊ Dic5
HallC5×C7×C17 ⋊ C3×Dic3
HallC3×C5×ΓL(1,𝔽8) ⋊ C4×C17
HallC7×C17 ⋊ C3×Dic15
HallC5×C17 ⋊ C3⋊(ΓL(1,𝔽8)⋊C4)
HallC3×ΓL(1,𝔽8) ⋊ C17×Dic5
HallC5×C7 ⋊ C3×Dic3×C17
SylowC7 ⋊ C3×C17×Dic15
SylowC5 ⋊ C17×C3⋊(ΓL(1,𝔽8)⋊C4)

Subgroups

subgroupG/H
center, hypercenterC2×C17C3⋊(C5⋊AGL(1,𝔽7))
commutator, hypocenterC3×C5×C7C3×C4×C17
FrattiniC2C17×C3⋊(C5⋊AGL(1,𝔽7))
Fitting, socleC2×C3×C5×C7×C17C2×C3
prime radicalC2×C3×C5×C17×ΓL(1,𝔽8)C2
cosocleC2×C3×C5×C7C2×C3×C17
LCSG ▷ C3×C5×C7
UCSC1 ◁ C2×C17

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

1428 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 141);