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C22×#11071 #72316893

order1024
nilpotency class3
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C212

Aliases

direct productC22×#11071

Subgroups

subgroupG/H
centerC23×C4#120
commutatorC24C24×C4
Frattini, cosocleC23×C4C25
socleC24#386
prime radicalC27C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

29053 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#394249, C2×#72239562, C2×#72238752
maximal normal subgroup ofC22×#394249, C2×#72239562, C2×#72238752

GAP

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

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