Finite Group Databasebeta

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C23×C32×AGL(1,𝔽7) #72245606

order3024
automorphism?
homologyAbelianization (H1) = C24×C33 · Schur multiplier (H2) = C26×C33

Aliases

direct productC23×C32×AGL(1,𝔽7)

Splittings

Fitting, socleC23×C32×C7 ⋊ C2×C3
derived, SylowC7 ⋊ C24×C33
HallC32×ΓL(1,𝔽8) ⋊ C24
HallC23×D7 ⋊ C33

Subgroups

subgroupG/H
center, hypercenterC23×C32AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C24×C33
FrattiniC1G
Fitting, socleC23×C32×C7C2×C3
LCSG ▷ C7
UCSC1 ◁ C23×C32

3752 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 504

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC2×C32×D4×AGL(1,𝔽7), C23×C3⋊D3×AGL(1,𝔽7), C2×AGL(1,𝔽7)×#638, …
maximal normal subgroup ofC2×C32×D4×AGL(1,𝔽7), C23×C3⋊D3×AGL(1,𝔽7), C2×AGL(1,𝔽7)×#638, …
maximal quotient ofC2×C32×D4×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((5,6)(7,8,9)(13,14,17,18,16,19,15), (1,2)(3,4)(5,6)(10,11,12)(14,15)(16,18)(17,19), (3,4)(14,16,17)(15,18,19), (1,2));