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C2×D9×#611 #72268811

order2592
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC2×D9×#611

Splittings

FittingC23×C92 ⋊ C22
SylowC92 ⋊ C22×D4

Subgroups

subgroupG/H
centerC22C2×D92
commutator, cosocleC2×C92C24
FrattiniC2×C32C22×D32
FittingC23×C92C22
socleC22×C32C2×D32
hypercenterC23D92
hypocenterC92C22×D4
LCSG ▷ C2×C92 ▷ C92
UCSC1 ◁ C22 ◁ C23

1251 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 4 generators on 24 points:

(3,4,5,6)(8,10)(9,13)(11,14)(12,15)(16,23)(17,20)(19,22)(21,24);(1,2)(7,14,15,9,8,10,13,12,11);(4,6);(17,18)(19,20)(21,23)(22,24)

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D4×D92, C2×#72219950
maximal normal subgroup ofC2×D4×D92, C2×#72219950

GAP

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

gap> G := Group((3,4,5,6)(8,10)(9,13)(11,14)(12,15)(16,23)(17,20)(19,22)(21,24), (1,2)(7,14,15,9,8,10,13,12,11), (4,6), (17,18)(19,20)(21,23)(22,24));