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Cayley diagram

C2×2−1+4⋊C5 #66247

order320
automorphismAut(G) = C2×#66297 (order 640) · Inn(G) = C24⋊C5 · Out(G) = C2×C4
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×2−1+4⋊C5
unambiguous semidirect product(C2×2−1+4)⋊C5

Splittings

Fitting, SylowC2×2−1+4 ⋊ C5
derived2−1+4 ⋊ C2×C5

Subgroups

subgroupG/H
center, socle, hypercenterC22C24⋊C5
commutator, cosocle, hypocenter2−1+4C2×C5
FrattiniC2C2×C24⋊C5
FittingC2×2−1+4C5
LCSG ▷ 2−1+4
UCSC1 ◁ C22

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 34 points:

(1,2)(4,7,22,10,13)(5,11,26,15,19)(6,17,31,27,33)(9,12,24,34,18)(14,16,23,25,30)(20,21,28,29,32);(3,23,8,11)(4,24,12,7)(5,17,16,28)(6,27,21,32)(9,15,13,30)(10,20,18,33)(14,34,19,22)(25,31,26,29)

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC22⋊#66244, #13046152, C22⋊#66243, …
maximal subgroup of#10617016, C4×2−1+4⋊C5, C2×#66243, …
maximal normal subgroup of#10617016, C4×2−1+4⋊C5, C2×#66243, …
maximal quotient ofC4×2−1+4⋊C5, C2×#564⋊C5, #3531⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1585);