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C22×#35979 #72299778

order1024
nilpotency class2
automorphism?
homologyAbelianization (H1) = C27 · Schur multiplier (H2) = C219

Aliases

direct productC22×#35979

Subgroups

subgroupG/H
center, socleC25C25
commutator, Frattini, cosocleC23C27
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C23 ▷ C1

92585 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 7 generators on 24 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((6,8)(9,10,13,15)(11,16,12,14), (5,6)(7,8)(9,11,13,12)(10,14,15,16)(17,18,20,22)(19,21,23,24), (9,11,13,12)(10,14,15,16)(17,18)(19,21)(20,22)(23,24), (9,12,13,11)(10,14,15,16)(17,19,20,23)(18,21,22,24), (9,10)(11,14)(12,16)(13,15), (3,4), (1,2)(3,4));