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C2×#616973 #72333135

order1024
nilpotency class2
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C215

Aliases

direct productC2×#616973

Subgroups

subgroupG/H
center, socleC25C25
commutator, Frattini, cosocleC24C26
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

74851 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#72323404, C2×#72323410, C2×#72323419, …
maximal normal subgroup ofC2×#72323404, C2×#72323410, C2×#72323419, …
maximal quotient ofC2×#72298513, C2×#72334391, C2×#72298452

GAP

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

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