Finite Group Databasebeta

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C2×C22⋊C4×#121 #72298705

order1024
nilpotency class2
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C216

Aliases

direct productC2×C22⋊C4×#121

Subgroups

subgroupG/H
center, socleC25C25
commutatorC23C25×C4
Frattini, cosocleC24C26
prime radicalC24×#121C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

83529 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC2×#72233554, C2×#72233563, C2×#72233528, …
maximal normal subgroup ofC2×#72233554, C2×#72233563, C2×#72233528, …
maximal quotient ofC2×C22⋊C4×#392, C22×C22⋊C4×#121, C2×#72332558, …

GAP

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

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