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C4×ΓL(1,𝔽8)×C11⋊C5 #17987607

order4620
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC4×ΓL(1,𝔽8)×C11⋊C5
unambiguous semidirect product(C4×C7)⋊(C3×C11⋊C5)
unambiguous semidirect product(C4×C11)⋊(C5×ΓL(1,𝔽8))
unambiguous semidirect product(C4×C11×ΓL(1,𝔽8))⋊C5
unambiguous semidirect product(C4×C7×C11⋊C5)⋊C3

Splittings

Fitting, HallC4×C7×C11 ⋊ C3×C5
derived, HallC7×C11 ⋊ C3×C4×C5
HallC4×C7×C11⋊C5 ⋊ C3
HallC4×C11×ΓL(1,𝔽8) ⋊ C5
HallC7×C11⋊C5 ⋊ C3×C4
HallC11×ΓL(1,𝔽8) ⋊ C4×C5
HallC4×C11 ⋊ C5×ΓL(1,𝔽8)
HallC4×C7 ⋊ C3×C11⋊C5
SylowC11 ⋊ C4×C5×ΓL(1,𝔽8)
SylowC7 ⋊ C3×C4×C11⋊C5

Subgroups

subgroupG/H
center, hypercenterC4ΓL(1,𝔽8)×C11⋊C5
commutator, hypocenterC7×C11C3×C4×C5
FrattiniC2C2×ΓL(1,𝔽8)×C11⋊C5
FittingC4×C7×C11C3×C5
socle, cosocleC2×C7×C11C2×C3×C5
prime radicalC2×ΓL(1,𝔽8)×C11⋊C5C2
LCSG ▷ C7×C11
UCSC1 ◁ C4

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 308

A minimal faithful permutation representation — 2 generators on 22 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic3×ΓL(1,𝔽8)×C11⋊C5, C3×C4×ΓL(1,𝔽8)×C11⋊C5, Dic5×ΓL(1,𝔽8)×C11⋊C5, …
maximal normal subgroup ofC3×C4×ΓL(1,𝔽8)×C11⋊C5, C4×C5×ΓL(1,𝔽8)×C11⋊C5, C4×C7×ΓL(1,𝔽8)×C11⋊C5
maximal quotient ofC4×C11⋊C5×C7⋊C9, Dic3×ΓL(1,𝔽8)×C11⋊C5, C3×C4×ΓL(1,𝔽8)×C11⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4620, 4);