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AGL(2,𝔽4) #72210408

order2880
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C2

Aliases

affine general linearAGL(2,𝔽4)

Splittings

Fitting, socleC24 ⋊ C3×A5
derived#11757162 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, hypocenter#11757162C3
Fitting, socle, cosocleC24C3×A5
LCSG ▷ #11757162
UCSC1

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

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

Conjugacy classes & character table

19 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC3×AΓL(2,𝔽4), C2×AΓL(2,𝔽4), #72277485, …
maximal subgroup ofPGL(3,𝔽4), AΓL(2,𝔽4), #72240854, …
maximal normal subgroup ofAΓL(2,𝔽4), C3×AGL(2,𝔽4), C2×AGL(2,𝔽4)
maximal quotient ofC3×AGL(2,𝔽4), C2×AGL(2,𝔽4)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

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