Finite Group Databasebeta

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C5×AGL(1,𝔽7)×D77 #44568630

order32340
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×AGL(1,𝔽7)×D77
unambiguous semidirect productD7⋊(C3×C5×D77)
unambiguous semidirect product(C7×AGL(1,𝔽7))⋊(C5×D11)

Splittings

Fitting, socle, HallC5×C72×C11 ⋊ C22×C3
derived, HallC72×C11 ⋊ C22×C3×C5
HallC5×D7×D77 ⋊ C3
HallC5×C7×C11×ΓL(1,𝔽8) ⋊ C22
HallD7×D77 ⋊ C3×C5
HallC7×C11×ΓL(1,𝔽8) ⋊ C22×C5
HallC5×C7×ΓL(1,𝔽8) ⋊ C2×D11
HallC5×C72 ⋊ C2×C3×D11
HallC7×ΓL(1,𝔽8) ⋊ C2×C5×D11
HallC5×C11 ⋊ D7×AGL(1,𝔽7)
SylowC72 ⋊ C2×C3×C5×D11
SylowC11 ⋊ C5×D7×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC5AGL(1,𝔽7)×D77
commutator, cosocle, hypocenterC72×C11C22×C3×C5
FrattiniC1G
Fitting, socleC5×C72×C11C22×C3
LCSG ▷ C72×C11
UCSC1 ◁ C5

168 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 2310

A minimal faithful permutation representation — 2 generators on 30 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,18)(15,16,17)(20,21)(22,23)(24,25)(26,27,28,29,30);(1,2,4,6,8,10,11,9,7,5,3)(13,14)(15,16)(17,18)(19,20,22,24,25,23,21)

Conjugacy classes & character table

1400 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 332);