Finite Group Databasebeta

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C3×C7×C17×ΓL(1,𝔽8)⋊C4 #44472643

order29988
automorphism?
homologyAbelianization (H1) = C32×C4×C7×C17 · Schur multiplier (H2) = C3

Aliases

direct productC3×C7×C17×ΓL(1,𝔽8)⋊C4
unambiguous semidirect productΓL(1,𝔽8)⋊(C3×C4×C7×C17)
unambiguous semidirect productDic7⋊(C32×C7×C17)
unambiguous semidirect product(C3×Dic7)⋊(C3×C7×C17)
unambiguous semidirect product(C17×Dic7)⋊(C32×C7)
unambiguous semidirect product(C3×C17×Dic7)⋊(C3×C7)

Splittings

derivedC7 ⋊ C32×C4×C7×C17
HallC3×C7×C17×ΓL(1,𝔽8) ⋊ C4
HallC7×C17×Dic7 ⋊ C32
HallC72×C17 ⋊ C32×C4
HallC3×C7×ΓL(1,𝔽8) ⋊ C4×C17
HallC7×Dic7 ⋊ C32×C17
SylowC72 ⋊ C32×C4×C17

Subgroups

subgroupG/H
center, hypercenterC2×C3×C7×C17AGL(1,𝔽7)
commutator, hypocenterC7C32×C4×C7×C17
FrattiniC2C3×C7×C17×AGL(1,𝔽7)
Fitting, socleC2×C3×C72×C17C2×C3
prime radicalC2×C3×C7×C17×ΓL(1,𝔽8)C2
cosocleC2×C7C2×C32×C7×C17
LCSG ▷ C7
UCSC1 ◁ C2×C3×C7×C17

152 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 4284

Conjugacy classes & character table

4998 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29988, 284);