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C24×D3×ΓL(1,𝔽8) #72244159

order2016
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C210

Aliases

direct productC24×D3×ΓL(1,𝔽8)

Splittings

Fitting, socleC24×C3×C7 ⋊ C2×C3
derivedC3×C7 ⋊ C25×C3
HallC3×ΓL(1,𝔽8) ⋊ C25
SylowC7 ⋊ C24×C3×D3

Subgroups

subgroupG/H
center, hypercenterC24D3×ΓL(1,𝔽8)
commutator, cosocle, hypocenterC3×C7C25×C3
FrattiniC1G
Fitting, socleC24×C3×C7C2×C3
LCSG ▷ C3×C7
UCSC1 ◁ C24

3126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 336

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×ΓL(1,𝔽8)×#120, C24×D3×AGL(1,𝔽7), D3×#10620789, …
maximal normal subgroup ofD3×ΓL(1,𝔽8)×#120, C24×D3×AGL(1,𝔽7), D3×#10620789, …
maximal quotient ofD3×ΓL(1,𝔽8)×H(5,𝔽2), C24×C3⋊D3×ΓL(1,𝔽8), C23×D32×ΓL(1,𝔽8)

GAP

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

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