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C2×C17⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽7)) #44472727

order29988
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C17⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽7))
unambiguous semidirect product(C2×C17)⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽7))
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C17⋊AGL(1,𝔽7))
unambiguous semidirect product(C2×C17⋊#64424)⋊C3
unambiguous semidirect product(C2×C17⋊#64425)⋊C3
unambiguous semidirect product(C2×C17⋊(ΓL(1,𝔽8)⋊D7))⋊C3

Splittings

Fitting, socleC2×C72×C17 ⋊ C2×C32
derived, HallC72×C17 ⋊ C22×C32
HallC17×ΓL(1,𝔽8)2 ⋊ C22
HallC2×C7⋊D119 ⋊ C32
HallΓL(1,𝔽8)2 ⋊ C2×D17
SylowC72 ⋊ C2×C32×D17
SylowC17 ⋊ C2×ΓL(1,𝔽8)⋊AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC2C17⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽7))
commutator, cosocle, hypocenterC72×C17C22×C32
FrattiniC1G
Fitting, socleC2×C72×C17C2×C32
LCSG ▷ C72×C17
UCSC1 ◁ C2

280 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 33
transitive degreeμt(G) = 714

A minimal faithful permutation representation — 2 generators on 33 points:

(3,4)(5,6)(7,9)(11,14,15,12,13,16)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33);(1,2)(4,6,8)(5,7,9)(10,11,13,16,15,14,12)(17,18,20,22,24,26,28,30,32,33,31,29,27,25,23,21,19)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29988, 368);