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C32×5−1+2 #11765511

order1125
nilpotency class2
isoclinismisoclinic class of H(3,𝔽5)
automorphism?
homologyAbelianization (H1) = C32×C52 · Schur multiplier (H2) = C3

Aliases

direct productC32×5−1+2
unambiguous semidirect productC25⋊(C32×C5)
unambiguous semidirect product(C3×C25)⋊(C3×C5)
unambiguous semidirect product(C32×C25)⋊C5

Subgroups

subgroupG/H
center, socleC32×C5C52
commutator, Frattini, cosocleC5C32×C52
prime radicalC32×C52C5
LCSG ▷ C5 ▷ C1
UCSC1 ◁ C32×C5 ◁ G

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 225

A minimal faithful permutation representation — 2 generators on 31 points:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup ofC32×D25⋊C5, C3⋊D3×5−1+2, #72241624, …
maximal subgroup ofC32×D25⋊C5, C3⋊D3×5−1+2, #72241624, …
maximal normal subgroup ofC32×D25⋊C5, C3⋊D3×5−1+2, #72241624, …
maximal quotient of3−1+2×5−1+2

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1125, 10);