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C9×5−1+2 #11765507

order1125
nilpotency class2
isoclinismisoclinic class of H(3,𝔽5)
automorphism?
homologyAbelianization (H1) = C52×C9 · Schur multiplier (H2) = C1

Aliases

direct productC9×5−1+2
unambiguous semidirect productC25⋊(C5×C9)
unambiguous semidirect product(C9×C25)⋊C5

Subgroups

subgroupG/H
centerC5×C9C52
commutatorC5C52×C9
Frattini, socle, cosocleC3×C5C3×C52
prime radicalC3×C52C3×C5
LCSG ▷ C5 ▷ C1
UCSC1 ◁ C5×C9 ◁ G

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 225

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

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup ofC9×D25⋊C5, C9×AGL(1,ℤ/25), D9×5−1+2, …
maximal subgroup ofC9×D25⋊C5, 3−1+2×5−1+2, D9×5−1+2, …
maximal normal subgroup ofC9×D25⋊C5, 3−1+2×5−1+2, D9×5−1+2, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1125, 6);