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C22×#6342372 #72289369

order2048
nilpotency class2
automorphism?
homologyAbelianization (H1) = C27 · Schur multiplier (H2) = C224

Aliases

direct productC22×#6342372

Subgroups

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

8403024 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)(14,16)(17,18,19,20);(13,14)(15,16);(17,18)(19,20);(9,10)(11,12)(13,14)(15,16)(17,18)(19,20);(5,6,7,8);(3,4);(1,2)(3,4)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC22×D4×#2846, C2×#72286609, C2×#72296173, …
maximal normal subgroup ofC22×D4×#2846, C2×#72286609, C2×#72296173, …
maximal quotient ofC2×#72286609, C22×#72286393

GAP

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

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