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C2×#120×#121 #72320415

order2048
nilpotency class2
automorphism?
homologyAbelianization (H1) = C27 · Schur multiplier (H2) = C222

Aliases

direct productC2×#120×#121

Subgroups

subgroupG/H
center, socleC25C26
commutator, Frattini, cosocleC24C27
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

1072920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 18 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC22×#120×#121, C2×D42×#120, C2×#72335129, …
maximal normal subgroup ofC22×#120×#121, C2×D42×#120, C2×#72335129, …
maximal quotient ofC2×#120×#392, C22×#120×#121

GAP

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

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