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C11×ΓL(1,𝔽8)⋊Dic5 #17987635

order4620
isoclinismisoclinic class of C5⋊AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C3×C4×C11 · Schur multiplier (H2) = C1

Aliases

direct productC11×ΓL(1,𝔽8)⋊Dic5
unambiguous semidirect productΓL(1,𝔽8)⋊(C11×Dic5)
unambiguous semidirect productDic35⋊(C3×C11)
unambiguous semidirect product(C11×Dic35)⋊C3

Splittings

derived, HallC5×C7 ⋊ C3×C4×C11
HallC11×Dic35 ⋊ C3
HallC5×C11×ΓL(1,𝔽8) ⋊ C4
HallC5×C7×C11 ⋊ C3×C4
HallC11×ΓL(1,𝔽8) ⋊ Dic5
HallDic35 ⋊ C3×C11
HallC5×ΓL(1,𝔽8) ⋊ C4×C11
HallC7×C11 ⋊ C3×Dic5
HallC5×C11 ⋊ ΓL(1,𝔽8)⋊C4
HallΓL(1,𝔽8) ⋊ C11×Dic5
SylowC7 ⋊ C3×C11×Dic5
SylowC5 ⋊ C11×ΓL(1,𝔽8)⋊C4

Subgroups

subgroupG/H
center, hypercenterC2×C11C5⋊AGL(1,𝔽7)
commutator, hypocenterC5×C7C3×C4×C11
FrattiniC2C11×C5⋊AGL(1,𝔽7)
Fitting, socleC2×C5×C7×C11C2×C3
prime radicalC2×C5×C11×ΓL(1,𝔽8)C2
cosocleC2×C5×C7C2×C3×C11
LCSG ▷ C5×C7
UCSC1 ◁ C2×C11

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 1540

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC3×C11×ΓL(1,𝔽8)⋊Dic5, C11×C3⋊(ΓL(1,𝔽8)⋊Dic5), C11×ΓL(1,𝔽8)⋊Dic25, …
maximal normal subgroup ofC3×C11×ΓL(1,𝔽8)⋊Dic5, C11⋊C5×ΓL(1,𝔽8)⋊Dic5, C5×C11×ΓL(1,𝔽8)⋊Dic5, …
maximal quotient ofC11×(C7⋊C9)⋊Dic5, C3×C11×ΓL(1,𝔽8)⋊Dic5, C11×C3⋊(ΓL(1,𝔽8)⋊Dic5), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4620, 32);