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C42×5−1+2 #14522249

order2000
nilpotency class2
isoclinismisoclinic class of H(3,𝔽5)
automorphism?
homologyAbelianization (H1) = C42×C52 · Schur multiplier (H2) = C4

Aliases

direct productC42×5−1+2
unambiguous semidirect product(C42×C25)⋊C5

Subgroups

subgroupG/H
centerC42×C5C52
commutatorC5C42×C52
Frattini, socle, cosocleC22×C5C22×C52
prime radicalC22×C52C22×C5
LCSG ▷ C5 ▷ C1
UCSC1 ◁ C42×C5 ◁ G

150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 400

A minimal faithful permutation representation — 2 generators on 33 points:

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

Conjugacy classes & character table

464 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC42×D25⋊C5, C42×AGL(1,ℤ/25), #72275631, …
maximal subgroup ofC42×D25⋊C5, #127×5−1+2, #72275631, …
maximal normal subgroup ofC42×D25⋊C5, #127×5−1+2, #72275631, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2000, 212);