Finite Group Databasebeta

← Prev (#63143) Next (#63145) →

Cayley diagram

C11×SL(2,𝔽3) #63144

order264
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C2×C5×S4 (order 240) · Inn(G) = A4 · Out(G) = C22×C5
homologyAbelianization (H1) = C3×C11 · Schur multiplier (H2) = C1

Aliases

direct productC11×SL(2,𝔽3)
unambiguous semidirect productQ8⋊(C3×C11)
unambiguous semidirect product(Q8×C11)⋊C3

Splittings

Fitting, HallQ8×C11 ⋊ C3
derived, SylowQ8 ⋊ C3×C11

Subgroups

subgroupG/H
center, socle, hypercenterC2×C11A4
commutator, cosocle, hypocenterQ8C3×C11
FrattiniC2C11×A4
FittingQ8×C11C3
LCSG ▷ Q8
UCSC1 ◁ C2×C11

14 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 88

A minimal faithful permutation representation — 2 generators on 19 points:

(2,3,5)(6,8,7)(9,10,11,12,13,14,15,16,17,18,19);(1,5,4,7)(2,8,6,3)

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC11⋊#226, C11⋊GL(2,𝔽3), C11⋊#859, …
maximal subgroup ofC11×#226, C11×GL(2,𝔽3), C11⋊#226, …
maximal normal subgroup ofC11×#226, C11×GL(2,𝔽3), C11⋊#226, …
maximal quotient ofC2×C11×SL(2,𝔽3), C11×Q8⋊C9, C3×C11×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(264, 12);