Finite Group Databasebeta

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C2×#72212205 #72267027

order2048
nilpotency class3
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C220

Aliases

direct productC2×#72212205

Subgroups

subgroupG/H
center, socleC24C22×#120
commutator, Frattini, cosocleC23×C4C26
prime radicalC25×2−1+4C2
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C26×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

4216016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

(9,10,13,15)(11,16,19,23)(12,14,21,17)(18,24,20,22);(1,2,3,4)(5,6)(7,8)(10,14,15,17)(16,24,23,22);(6,8)(10,15)(14,17)(16,23)(22,24);(2,4);(9,11,13,19)(10,16,15,23)(12,20,21,18)(14,22,17,24);(25,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of#72267040, C2×#72212713, C2×#72212876, …
maximal normal subgroup of#72267040, #72267488

GAP

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

gap> G := Group((9,10,13,15)(11,16,19,23)(12,14,21,17)(18,24,20,22), (1,2,3,4)(5,6)(7,8)(10,14,15,17)(16,24,23,22), (6,8)(10,15)(14,17)(16,23)(22,24), (2,4), (9,11,13,19)(10,16,15,23)(12,20,21,18)(14,22,17,24), (25,26));