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C2×C3×#70918 #72227375

order2304
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25

Aliases

direct productC2×C3×#70918

Splittings

FittingC2×C32×#442 ⋊ C2
SylowC32 ⋊ C2×AGL(1,ℤ/16)

Subgroups

subgroupG/H
centerC22×C3#5014
commutatorC3×C8C23×C3×C4
FrattiniC2×C8C23×C3×D3
FittingC2×C32×#442C2
socleC22×C32#442
prime radicalC2×C3×#5147C2
cosocleC2×C3×C8C24×C3
hypercenterC2×C3×#442D3
hypocenterC3C2×C3×AGL(1,ℤ/16)
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#97 ◁ C2×C3×#442

1724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 96

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC3×#72228227
maximal normal subgroup ofC3×#72228227

GAP

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

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