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H(5,𝔽2)×C11⋊C5 #13183327

order1760
automorphism?
homologyAbelianization (H1) = C24×C5 · Schur multiplier (H2) = C25

Aliases

direct productH(5,𝔽2)×C11⋊C5
unambiguous semidirect product(C11×H(5,𝔽2))⋊C5

Splittings

Fitting, HallC11×H(5,𝔽2) ⋊ C5
SylowC11 ⋊ C5×H(5,𝔽2)

Subgroups

subgroupG/H
center, FrattiniC2C24×C11⋊C5
commutator, socle, cosocleC2×C11C24×C5
FittingC11×H(5,𝔽2)C5
hypercenterH(5,𝔽2)C11⋊C5
hypocenterC11C5×H(5,𝔽2)
LCSG ▷ C2×C11 ▷ C11
UCSC1 ◁ C2 ◁ H(5,𝔽2)

332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofH(5,𝔽2)×AGL(1,𝔽11), C11⋊C5×B(4,𝔽2), C11⋊C5×#453, …
maximal normal subgroup ofH(5,𝔽2)×AGL(1,𝔽11), C11⋊C5×B(4,𝔽2), C11⋊C5×#453, …
maximal quotient of#72222614, C2×H(5,𝔽2)×C11⋊C5, C11⋊C5×#535, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1760, 1136);