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C22⋊C4×5−1+2 #14522252

order2000
nilpotency class2
isoclinismisoclinic class of D4×H(3,𝔽5)
automorphism?
homologyAbelianization (H1) = C2×C4×C52 · Schur multiplier (H2) = C22

Aliases

direct productC22⋊C4×5−1+2
unambiguous semidirect productC22⋊(C4×5−1+2)
unambiguous semidirect product(C22⋊C4×C25)⋊C5

Subgroups

subgroupG/H
center, Frattini, socle, cosocleC22×C5C22×C52
commutatorC2×C5C2×C4×C52
prime radicalC23×C52C2×C5
LCSG ▷ C2×C5 ▷ C1
UCSC1 ◁ C22×C5 ◁ G

170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 200

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

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

Conjugacy classes & character table

290 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC22⋊C4×D25⋊C5, C22⋊C4×AGL(1,ℤ/25), #72275639, …
maximal subgroup ofC22⋊C4×D25⋊C5, #120×5−1+2, #72275639, …
maximal normal subgroup ofC22⋊C4×D25⋊C5, #120×5−1+2, #72275639, …
maximal quotient of#100×5−1+2, #99×5−1+2, #104×5−1+2

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2000, 215);