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C11×ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4) #34084746

order19404
automorphism?
homologyAbelianization (H1) = C32×C4×C11 · Schur multiplier (H2) = C3

Aliases

direct productC11×ΓL(1,𝔽8)⋊(ΓL(1,𝔽8)⋊C4)
unambiguous semidirect productΓL(1,𝔽8)⋊(C11×ΓL(1,𝔽8)⋊C4)
unambiguous semidirect product(ΓL(1,𝔽8)⋊Dic7)⋊(C3×C11)
unambiguous semidirect product#10596874⋊(C3×C11)
unambiguous semidirect product#10596875⋊(C3×C11)
unambiguous semidirect product(C11×#10596874)⋊C3
unambiguous semidirect product(C11×#10596875)⋊C3
unambiguous semidirect product(C11×ΓL(1,𝔽8)⋊Dic7)⋊C3

Splittings

derived, SylowC72 ⋊ C32×C4×C11
HallC11×ΓL(1,𝔽8)2 ⋊ C4
HallC11×C7⋊Dic7 ⋊ C32
HallC72×C11 ⋊ C32×C4
HallΓL(1,𝔽8)2 ⋊ C4×C11
HallC7⋊Dic7 ⋊ C32×C11

Subgroups

subgroupG/H
center, hypercenterC2×C11ΓL(1,𝔽8)⋊AGL(1,𝔽7)
commutator, hypocenterC72C32×C4×C11
FrattiniC2C11×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
Fitting, socleC2×C72×C11C2×C32
prime radicalC2×C11×ΓL(1,𝔽8)2C2
cosocleC2×C72C2×C32×C11
LCSG ▷ C72
UCSC1 ◁ C2×C11

168 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 924

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19404, 237);