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AΓL(2,𝔽4) #72215738

order5760
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C2

Aliases

affine general semilinearAΓL(2,𝔽4)

Splittings

Fitting, socleC24 ⋊ ΓL(2,𝔽4)
derivedAGL(2,𝔽4) ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterAGL(2,𝔽4)C2
Fitting, socleC24ΓL(2,𝔽4)
LCSG ▷ AGL(2,𝔽4)
UCSC1

199 conjugacy classes of subgroups; only the 6 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 6 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Permutation generators:

(1,6)(2,11)(3,12)(4,14)(7,25)(8,31)(9,24)(10,23)(13,18)(15,17)(16,37)(21,27)(22,29)(26,39)(28,38)(30,43)(32,42)(33,41)(34,40)(35,44)(36,45)(46,47);(2,12,14)(3,11,4)(5,18,13)(7,26,10)(8,29,35)(9,28,27)(15,17,19)(21,38,24)(22,31,44)(23,39,25)(30,34,36)(40,43,45);(2,11)(3,13)(4,15)(5,19)(7,27)(8,26)(9,30)(10,34)(12,18)(14,17)(21,25)(22,38)(23,40)(24,43)(28,29)(31,39)(35,36)(44,45);(2,12)(3,11)(4,16)(5,20)(7,26)(8,29)(9,33)(10,32)(13,18)(14,37)(21,38)(22,31)(23,42)(24,41)(25,39)(27,28)(35,47)(44,46);(4,16,11)(5,20,13)(9,33,27)(10,32,26)(12,14,37)(15,19,17)(23,42,25)(24,41,38)(29,35,47)(30,36,34)(31,44,46)(40,45,43);(4,17,20)(5,15,37)(9,34,47)(10,36,33)(11,13,12)(14,19,16)(23,43,46)(24,45,42)(25,31,38)(26,27,29)(30,32,35)(40,41,44);(1,7)(6,21)(8,28)(9,29)(10,30)(22,39)(23,31)(24,40)(25,46)(26,34)(27,47)(32,36)(33,35)(38,43)(41,45)(42,44);(1,8)(6,22)(7,28)(9,32)(10,27)(21,39)(23,41)(24,25)(26,33)(29,36)(30,47)(31,45)(34,35)(38,42)(40,46)(43,44);(1,9)(6,23)(7,29)(8,32)(10,35)(21,31)(22,41)(24,44)(25,43)(26,47)(27,34)(28,36)(30,33)(38,46)(39,45)(40,42);(1,10)(6,24)(7,30)(8,27)(9,35)(21,40)(22,25)(23,44)(26,36)(28,47)(29,33)(31,42)(32,34)(38,45)(39,46)(41,43)

Conjugacy classes & character table

17 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240863, C3×AΓL(2,𝔽4), #72241556, …
maximal normal subgroup ofC3×AΓL(2,𝔽4), C2×AΓL(2,𝔽4)
maximal quotient ofC3×AΓL(2,𝔽4), #72241556, C2×AΓL(2,𝔽4), …
automorphism group ofC22⋊A4, C2×C22⋊A4

GAP

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

gap> G := Group((1,6)(2,11)(3,12)(4,14)(7,25)(8,31)(9,24)(10,23)(13,18)(15,17)(16,37)(21,27)(22,29)(26,39)(28,38)(30,43)(32,42)(33,41)(34,40)(35,44)(36,45)(46,47), (2,12,14)(3,11,4)(5,18,13)(7,26,10)(8,29,35)(9,28,27)(15,17,19)(21,38,24)(22,31,44)(23,39,25)(30,34,36)(40,43,45), (2,11)(3,13)(4,15)(5,19)(7,27)(8,26)(9,30)(10,34)(12,18)(14,17)(21,25)(22,38)(23,40)(24,43)(28,29)(31,39)(35,36)(44,45), (2,12)(3,11)(4,16)(5,20)(7,26)(8,29)(9,33)(10,32)(13,18)(14,37)(21,38)(22,31)(23,42)(24,41)(25,39)(27,28)(35,47)(44,46), (4,16,11)(5,20,13)(9,33,27)(10,32,26)(12,14,37)(15,19,17)(23,42,25)(24,41,38)(29,35,47)(30,36,34)(31,44,46)(40,45,43), (4,17,20)(5,15,37)(9,34,47)(10,36,33)(11,13,12)(14,19,16)(23,43,46)(24,45,42)(25,31,38)(26,27,29)(30,32,35)(40,41,44), (1,7)(6,21)(8,28)(9,29)(10,30)(22,39)(23,31)(24,40)(25,46)(26,34)(27,47)(32,36)(33,35)(38,43)(41,45)(42,44), (1,8)(6,22)(7,28)(9,32)(10,27)(21,39)(23,41)(24,25)(26,33)(29,36)(30,47)(31,45)(34,35)(38,42)(40,46)(43,44), (1,9)(6,23)(7,29)(8,32)(10,35)(21,31)(22,41)(24,44)(25,43)(26,47)(27,34)(28,36)(30,33)(38,46)(39,45)(40,42), (1,10)(6,24)(7,30)(8,27)(9,35)(21,40)(22,25)(23,44)(26,36)(28,47)(29,33)(31,42)(32,34)(38,45)(39,46)(41,43));