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ΓL(1,𝔽8)⋊D371 #29510048

order15582
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊D371
unambiguous semidirect productC53⋊(ΓL(1,𝔽8)⋊D7)

Splittings

Fitting, socle, derived, HallC72×C53 ⋊ C2×C3
HallC7×ΓL(1,𝔽8)×C53 ⋊ C2
HallC7⋊D371 ⋊ C3
HallC7×ΓL(1,𝔽8) ⋊ D53
SylowC53 ⋊ ΓL(1,𝔽8)⋊D7
SylowC72 ⋊ C3×D53

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC72×C53C2×C3
LCSG ▷ C72×C53
UCSC1

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 67
transitive degreeμt(G) = 1113

A minimal faithful permutation representation — 2 generators on 67 points:

(2,3)(4,5)(6,7)(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)(55,58,59,56,57,60)(62,63)(64,65)(66,67);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,55,57,60,59,58,56)(61,62,64,66,67,65,63)

Polycyclic presentation:

2,3,7^2,53|c^a=c^6,d^a=d^6,e^a=e^52,c^b=c^2

Conjugacy classes & character table

932 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD53×ΓL(1,𝔽8)⋊D7, D7×ΓL(1,𝔽8)⋊D53, AGL(1,𝔽7)×D371, …
maximal normal subgroup ofD53×ΓL(1,𝔽8)⋊D7, D7×ΓL(1,𝔽8)⋊D53, AGL(1,𝔽7)×D371, …
maximal quotient ofΓL(1,𝔽8)⋊Dic371, C2×ΓL(1,𝔽8)⋊D371

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15582, 26);