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ΓL(1,𝔽8)×D79 #15693297

order3318
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×D79

Splittings

Fitting, socle, derived, HallC7×C79 ⋊ C2×C3
HallΓL(1,𝔽8)×C79 ⋊ C2
HallC7×D79 ⋊ C3
SylowC79 ⋊ C2×ΓL(1,𝔽8)
SylowC7 ⋊ C3×D79

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C79C2×C3
LCSG ▷ C7×C79
UCSC1

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 86
transitive degreeμt(G) = 553

A minimal faithful permutation representation — 2 generators on 86 points:

(2,3,4)(5,7,6)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86);(1,2,3,5,4,6,7)(9,86)(10,85)(11,84)(12,83)(13,82)(14,81)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(21,74)(22,73)(23,72)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Conjugacy classes & character table

205 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×ΓL(1,𝔽8)×D79, ΓL(1,𝔽8)×D395, C49⋊C3×D79, …
maximal normal subgroup ofC5×ΓL(1,𝔽8)×D79, C7×ΓL(1,𝔽8)×D79, C11×ΓL(1,𝔽8)×D79
maximal quotient ofC5×ΓL(1,𝔽8)×D79, ΓL(1,𝔽8)×D395, C49⋊C3×D79, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3318, 14);