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H(5,𝔽2)×C13⋊C3 #11929250

order1248
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productH(5,𝔽2)×C13⋊C3

Splittings

Fitting, HallC13×H(5,𝔽2) ⋊ C3
SylowC13 ⋊ C3×H(5,𝔽2)

Subgroups

subgroupG/H
center, FrattiniC2C24×C13⋊C3
commutator, socle, cosocleC2×C13C24×C3
FittingC13×H(5,𝔽2)C3
hypercenterH(5,𝔽2)C13⋊C3
hypocenterC13C3×H(5,𝔽2)
LCSG ▷ C2×C13 ▷ C13
UCSC1 ◁ C2 ◁ H(5,𝔽2)

332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 104

A minimal faithful permutation representation — 4 generators on 21 points:

(1,2)(3,7)(4,5)(6,8)(10,11,13)(12,15,18)(14,17,16)(19,21,20);(2,5)(6,7)(9,10,12,11,14,18,15,19,21,13,16,20,17);(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)×D13⋊C3, C13⋊C3×B(4,𝔽2), C13⋊C3×#997, …
maximal normal subgroup ofH(5,𝔽2)×D13⋊C3, C13⋊C3×B(4,𝔽2), C13⋊C3×#997, …
maximal quotient of#72259622, C2×H(5,𝔽2)×C13⋊C3

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 1362);