Finite Group Databasebeta

← Prev (#72236724) Next (#72236726) →

C2×#6385983 #72236725

order1024
nilpotency class3
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C214

Aliases

direct productC2×#6385983

Subgroups

subgroupG/H
center, socleC23C23×C22⋊C4
commutatorC23C25×C4
Frattini, cosocleC22×C4C26
prime radicalC23×D42C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

63169 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 6 generators on 18 points:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72236751, #72240299, #72251097, …
maximal normal subgroup of#72236751, #72240299, #72251097, …
maximal quotient of#72251097

GAP

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

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