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C5⋊#6728 #11925268

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

Aliases

unambiguous semidirect productC5⋊#6728

Subgroups

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

46 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 53
transitive degreeμt(G) = 240

A minimal faithful permutation representation — 2 generators on 53 points:

(1,2,8,23,40,24,34,20,13,3)(4,17,5,21,10,26,47,36,14,18)(6,7)(9,27,15,11,25,45,35,16,12,28)(19,41,38,46,33,43,22,39,48,32)(29,30)(31,37)(42,44)(49,50,52,53,51);(1,4,19,42,41,36,24,26,43,44,22,5)(2,9,3,14,37,18,34,45,40,21,31,10)(6,23,27,48,28,20,7,13,35,46,25,8)(11,32,47,39,16,30,12,33,17,38,15,29)(49,50)(51,52)

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72277246, #72277296, #72277300
maximal normal subgroup of#72277246, #72277296, #72277300

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1200, 478);