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C24×C3×AGL(1,𝔽7) #72244134

order2016
automorphism?
homologyAbelianization (H1) = C25×C32 · Schur multiplier (H2) = C210×C3

Aliases

direct productC24×C3×AGL(1,𝔽7)

Splittings

Fitting, socleC24×C3×C7 ⋊ C2×C3
derived, SylowC7 ⋊ C25×C32
HallC24×D7 ⋊ C32
HallC3×ΓL(1,𝔽8) ⋊ C25

Subgroups

subgroupG/H
center, hypercenterC24×C3AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C25×C32
FrattiniC1G
Fitting, socleC24×C3×C7C2×C3
LCSG ▷ C7
UCSC1 ◁ C24×C3

4488 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)(12,13,16,17,15,18,14);(5,6)(13,15,16)(14,17,18);(3,4);(1,2)(3,4)(5,6)(7,8)(13,14)(15,17)(16,18);(1,2)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC3×#120×AGL(1,𝔽7), C24×D3×AGL(1,𝔽7), AGL(1,𝔽7)×#953, …
maximal normal subgroup ofC3×#120×AGL(1,𝔽7), C24×D3×AGL(1,𝔽7), AGL(1,𝔽7)×#953, …
maximal quotient ofC3×H(5,𝔽2)×AGL(1,𝔽7), C24×ΓL(1,𝔽8)⋊AGL(1,𝔽7), C24×C32×AGL(1,𝔽7)

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