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C22×#759288 #72289173

order2048
nilpotency class2
automorphism?
homologyAbelianization (H1) = C27 · Schur multiplier (H2) = C223

Aliases

direct productC22×#759288

Subgroups

subgroupG/H
center, socleC26C25
commutator, Frattini, cosocleC24C27
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C26 ◁ G
LECSG ▷ C24 ▷ C1

3389304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 7 generators on 20 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC22×#72232804, C22×#72231456, C2×#72287912, …
maximal normal subgroup ofC22×#72232804, C22×#72231456, C2×#72287912, …
maximal quotient ofC2×#72287912

GAP

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

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