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C5⋊#6729 #11925267

order1200
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊#6729

Splittings

derivedC5×SL(2,𝔽5) ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC2C5⋊S5
commutator, cosocle, hypocenterC5×SL(2,𝔽5)C2
Fitting, socleC2×C5S5
LCSG ▷ C5×SL(2,𝔽5)
UCSC1 ◁ C2

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 200

A minimal faithful permutation representation — 2 generators on 45 points:

(1,5,16,23,9)(2,6,15,22,10)(3,7,14,21,11)(4,8,13,24,12)(17,28,31,37,34)(18,27,32,38,33)(19,26,29,39,36)(20,25,30,40,35)(41,42,44,45,43);(1,13,33,19,3,15,35,17)(2,14,34,20,4,16,36,18)(5,21,29,9,7,23,31,11)(6,22,32,12,8,24,30,10)(25,38,26,39,27,40,28,37)(41,43)(42,45)

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72277234, #72277296, #72277297
maximal normal subgroup of#72277234, #72277296, #72277297

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1200, 477);