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ΓL(1,𝔽8)×D299 #27160207

order12558
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×D299
unambiguous semidirect productC23⋊(ΓL(1,𝔽8)×D13)

Splittings

Fitting, socle, derived, HallC7×C13×C23 ⋊ C2×C3
HallC13×ΓL(1,𝔽8)×C23 ⋊ C2
HallC7×D299 ⋊ C3
HallΓL(1,𝔽8)×C23 ⋊ D13
HallC13×C23 ⋊ C2×ΓL(1,𝔽8)
HallC13×ΓL(1,𝔽8) ⋊ D23
HallC7×C23 ⋊ C3×D13
HallC7×C13 ⋊ C3×D23
SylowC23 ⋊ ΓL(1,𝔽8)×D13
SylowC13 ⋊ ΓL(1,𝔽8)×D23
SylowC7 ⋊ C3×D299

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C13×C23C2×C3
LCSG ▷ C7×C13×C23
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 43
transitive degreeμt(G) = 2093

A minimal faithful permutation representation — 2 generators on 43 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofAGL(1,𝔽7)×D299, (ΓL(1,𝔽8)⋊D13)⋊D23, ΓL(1,𝔽8)×D13×D23, …
maximal normal subgroup ofAGL(1,𝔽7)×D299, (ΓL(1,𝔽8)⋊D13)⋊D23, ΓL(1,𝔽8)×D13×D23, …
maximal quotient ofΓL(1,𝔽8)×Dic299, C2×ΓL(1,𝔽8)×D299, C7⋊C9×D299, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12558, 30);