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Cayley diagram

D5×H(5,𝔽2) #66284

order320
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productD5×H(5,𝔽2)

Splittings

FittingC5×H(5,𝔽2) ⋊ C2
SylowC5 ⋊ C2×H(5,𝔽2)

Subgroups

subgroupG/H
center, FrattiniC2C24×D5
commutator, socle, cosocleC2×C5C25
FittingC5×H(5,𝔽2)C2
hypercenterH(5,𝔽2)D5
hypocenterC5C2×H(5,𝔽2)
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ H(5,𝔽2)

898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 40

A minimal faithful permutation representation — 5 generators on 13 points:

(2,6)(5,7)(9,10,12,13,11);(3,8)(5,7);(1,3)(2,5)(4,8)(6,7);(1,2)(3,7)(4,6)(5,8);(10,11)(12,13)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10605015, #10605017, D5×#453, …
maximal normal subgroup of#10605015, #10605017, D5×#453, …
maximal quotient of#10617758, #10617761, #10617762, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1622);