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C11×C3⋊#170 #13050339

order1320
isoclinismisoclinic class of ΓL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C8×C11 · Schur multiplier (H2) = C1

Aliases

direct productC11×C3⋊#170
unambiguous semidirect productC3⋊(C11×#170)

Splittings

derived, HallC3×C5 ⋊ C8×C11
HallC3×C5×C11 ⋊ C8
HallC5×C11 ⋊ C3⋊C8
HallC3×C11 ⋊ #170
SylowC5 ⋊ C11×C3⋊C8
SylowC3 ⋊ C11×#170

Subgroups

subgroupG/H
center, hypercenterC2×C11ΓL(1,𝔽16)
commutator, hypocenterC3×C5C8×C11
FrattiniC2C11×ΓL(1,𝔽16)
Fitting, socle, prime radicalC2×C3×C5×C11C4
cosocleC3×Dic5C2×C11
LCSG ▷ C3×C5
UCSC1 ◁ C2×C11

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 1320

A minimal faithful permutation representation — 2 generators on 27 points:

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

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1320, 43);