Finite Group Databasebeta

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C23×#2010 #72239237

order1024
nilpotency class3
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C216

Aliases

direct productC23×#2010

Subgroups

subgroupG/H
centerC23×C4#127
commutator, Frattini, cosocleC22×C4C26
socleC24#392
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

82245 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239240, C23×#23882, C23×#23897, …
maximal normal subgroup of#72239240, C23×#23882, C23×#23897, …
maximal quotient of#72249657, #72257052, C22×#233610, …

GAP

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

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