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C7×C52⋊H(3,𝔽5) #36346234

order21875
nilpotency class2
automorphism?
homologyAbelianization (H1) = C53×C7 · Schur multiplier (H2) = C56

Aliases

direct productC7×C52⋊H(3,𝔽5)
unambiguous semidirect productC52⋊(C7×H(3,𝔽5))
unambiguous semidirect product(C52×C7)⋊H(3,𝔽5)

Subgroups

subgroupG/H
center, socleC52×C7C53
commutator, Frattini, cosocleC52C53×C7
LCSG ▷ C52 ▷ C1
UCSC1 ◁ C52×C7 ◁ G

1282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 57
transitive degreeμt(G) = 875

A minimal faithful permutation representation — 3 generators on 57 points:

(1,2,4,8,15)(3,6,10,17,22)(5,11,20,25,21)(7,13,19,16,23)(9,18,24,14,12)(26,27,29,33,40)(28,31,35,42,47)(30,36,45,50,46)(32,38,44,41,48)(34,43,49,39,37)(51,52,53,54,55,56,57);(2,5,12,13,6)(4,9,10,11,19)(8,16,20,17,18)(15,22,23,24,25);(27,30,37,38,31)(29,34,35,36,44)(33,41,45,42,43)(40,47,48,49,50)

Conjugacy classes & character table

1715 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21875, 45);