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C2×#674141 #72299350

order1024
nilpotency class2
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C216

Aliases

direct productC2×#674141

Subgroups

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

82185 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 256

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

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#72297600, C2×#72297599, C2×#72261412, …
maximal normal subgroup ofC2×#72297600, C2×#72297599, C2×#72261412, …
maximal quotient ofC22×#674141, C2×#72298133, C2×#72334399, …

GAP

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

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