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C43×H(3,𝔽5)⋊C5 #41001835

order26875
nilpotency class3
automorphism?
homologyAbelianization (H1) = C52×C43 · Schur multiplier (H2) = C52

Aliases

direct productC43×H(3,𝔽5)⋊C5
unambiguous semidirect productH(3,𝔽5)⋊(C5×C43)
unambiguous semidirect product(C43×H(3,𝔽5))⋊C5

Subgroups

subgroupG/H
center, socleC5×C43H(3,𝔽5)
commutator, Frattini, cosocleC52C52×C43
LCSG ▷ C52 ▷ C5 ▷ C1
UCSC1 ◁ C5×C43 ◁ C52×C43 ◁ G

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 68
transitive degreeμt(G) = 1075

A minimal faithful permutation representation — 2 generators on 68 points:

(1,2,4,8,15)(3,5,10,16,22)(6,12,20,25,21)(7,11,19,17,23)(9,18,24,14,13)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68);(2,5,11,13,6)(4,9,10,12,19)(8,16,17,18,20)

Conjugacy classes & character table

2107 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26875, 7);