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D19×ΓL(1,𝔽8)⋊AGL(1,𝔽7) #45692029

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

Aliases

direct productD19×ΓL(1,𝔽8)⋊AGL(1,𝔽7)

Splittings

Fitting, socle, derived, HallC72×C19 ⋊ C22×C32
HallC19×ΓL(1,𝔽8)2 ⋊ C22
HallD19×C7⋊D7 ⋊ C32
HallΓL(1,𝔽8)2 ⋊ C2×D19
SylowC72 ⋊ C2×C32×D19
SylowC19 ⋊ C2×ΓL(1,𝔽8)⋊AGL(1,𝔽7)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC72×C19C22×C32
LCSG ▷ C72×C19
UCSC1

280 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 399

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33516, 827);