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C2×ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊D19) #45692157

order33516
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊D19)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(ΓL(1,𝔽8)⋊D19)

Splittings

Fitting, socleC2×C72×C19 ⋊ C2×C32
derived, HallC72×C19 ⋊ C22×C32
HallC19×ΓL(1,𝔽8)2 ⋊ C22
HallC2×C7⋊D133 ⋊ C32
HallΓL(1,𝔽8)2 ⋊ C2×D19
SylowC72 ⋊ C2×C32×D19
SylowC19 ⋊ C2×ΓL(1,𝔽8)⋊AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊D19)
commutator, cosocle, hypocenterC72×C19C22×C32
FrattiniC1G
Fitting, socleC2×C72×C19C2×C32
LCSG ▷ C72×C19
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) = 35
transitive degreeμt(G) = 798

A minimal faithful permutation representation — 2 generators on 35 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)(34,35);(1,2)(4,6,8)(5,7,9)(10,11,13,16,15,14,12)(17,18,20,22,24,26,28,30,32,34,35,33,31,29,27,25,23,21,19)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33516, 955);