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C11⋊(ΓL(1,𝔽8)⋊C4) #11745102

order924
isoclinismisoclinic class of C11⋊AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC11⋊(ΓL(1,𝔽8)⋊C4)
unambiguous semidirect productΓL(1,𝔽8)⋊Dic11
unambiguous semidirect productDic77⋊C3

Splittings

derived, HallC7×C11 ⋊ C3×C4
HallDic77 ⋊ C3
HallC11×ΓL(1,𝔽8) ⋊ C4
HallΓL(1,𝔽8) ⋊ Dic11
SylowC11 ⋊ ΓL(1,𝔽8)⋊C4
SylowC7 ⋊ C3×Dic11

Subgroups

subgroupG/H
center, Frattini, hypercenterC2C11⋊AGL(1,𝔽7)
commutator, hypocenterC7×C11C3×C4
Fitting, socle, cosocleC2×C7×C11C2×C3
prime radicalC2×C11×ΓL(1,𝔽8)C2
LCSG ▷ C7×C11
UCSC1 ◁ C2

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 308

Polycyclic presentation:

2,3,2,7,11|a^2=c,d^a=d^6,e^a=e^10,d^b=d^2

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD11×ΓL(1,𝔽8)⋊C4, AGL(1,𝔽7)×Dic11, #10597505⋊C3, …
maximal normal subgroup ofD11×ΓL(1,𝔽8)⋊C4, AGL(1,𝔽7)×Dic11, #10597505⋊C3, …
maximal quotient ofC11⋊(ΓL(1,𝔽8)⋊C8), C2×C11⋊(ΓL(1,𝔽8)⋊C4), C11⋊(Dic7⋊C9), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(924, 3);