Finite Group Databasebeta

← Prev (#10621715) Next (#10621717) →

C52×3−1+2 #10621716

order675
nilpotency class2
isoclinismisoclinic class of H(3,𝔽3)
automorphism?
homologyAbelianization (H1) = C32×C52 · Schur multiplier (H2) = C5

Aliases

direct productC52×3−1+2
unambiguous semidirect productC9⋊(C3×C52)
unambiguous semidirect product(C5×C9)⋊(C3×C5)

Subgroups

subgroupG/H
center, socleC3×C52C32
commutator, Frattini, cosocleC3C32×C52
prime radicalC32×C52C3
LCSG ▷ C3 ▷ C1
UCSC1 ◁ C3×C52 ◁ G

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 225

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2,3,4,5)(11,12,14,13,16,18,17,15,19);(6,7,8,9,10)(12,15,16)(14,18,19)

Conjugacy classes & character table

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

Relationships

Fitting subgroup ofC5×D5×3−1+2, 3−1+2×C5⋊D5, C52×AGL(1,ℤ/9), …
maximal subgroup ofC5×D5×3−1+2, 3−1+2×C5⋊D5, C52×AGL(1,ℤ/9), …
maximal normal subgroup ofC5×D5×3−1+2, 3−1+2×C5⋊D5, C52×AGL(1,ℤ/9), …
maximal quotient ofC2×C52×3−1+2, 3−1+2×5−1+2

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(675, 15);