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C2×C3×#7389 #72255384

order1536
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C23

Aliases

direct productC2×C3×#7389

Subgroups

subgroupG/H
center, socleC22×C3#1333
commutatorC16C22×C3×C8
Frattini, cosocle#346C23×C3
prime radicalC2×C3×#508C4
LCSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ C2×C3×#346 ◁ G

500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 37 points:

(3,4,5,12,7,13,18,31,9,14,6,20,22,33,21,11,10,15,19,28,8,26,16,29,25,27,24,32,17,34,23,30)(35,36,37);(1,2);(4,11,27,29,15,30,14,31)(5,16,19,18)(6,21,24,23)(7,22)(8,17)(9,25)(12,26,20,33,28,13,32,34)

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×D3×#7389, C2×#72222517
maximal subgroup ofC2×D3×#7389, C2×C3×AGL(1,ℤ/32), C2×#72222517
maximal normal subgroup ofC2×D3×#7389, C2×C3×AGL(1,ℤ/32), C2×#72222517

GAP

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

gap> G := Group((3,4,5,12,7,13,18,31,9,14,6,20,22,33,21,11,10,15,19,28,8,26,16,29,25,27,24,32,17,34,23,30)(35,36,37), (1,2), (4,11,27,29,15,30,14,31)(5,16,19,18)(6,21,24,23)(7,22)(8,17)(9,25)(12,26,20,33,28,13,32,34));