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ΓL(1,𝔽8)×Dic47 #16841114

order3948
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×Dic47
unambiguous semidirect productC47⋊(C4×ΓL(1,𝔽8))
unambiguous semidirect product(C7×Dic47)⋊C3

Splittings

derived, HallC7×C47 ⋊ C3×C4
HallC7×Dic47 ⋊ C3
HallΓL(1,𝔽8)×C47 ⋊ C4
SylowC47 ⋊ C4×ΓL(1,𝔽8)
SylowC7 ⋊ C3×Dic47

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)×D47
commutator, hypocenterC7×C47C3×C4
Fitting, socle, cosocleC2×C7×C47C2×C3
prime radicalC2×ΓL(1,𝔽8)×C47C2
LCSG ▷ C7×C47
UCSC1 ◁ C2

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 58
transitive degreeμt(G) = 1316

A minimal faithful permutation representation — 2 generators on 58 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);(1,2,3,5,4,6,7)(8,10,9,11)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×ΓL(1,𝔽8)×Dic47, ΓL(1,𝔽8)×Dic141, C5×ΓL(1,𝔽8)×Dic47, …
maximal normal subgroup ofC3×ΓL(1,𝔽8)×Dic47, C5×ΓL(1,𝔽8)×Dic47, C7×ΓL(1,𝔽8)×Dic47
maximal quotient ofC7⋊C9×Dic47, C3×ΓL(1,𝔽8)×Dic47, ΓL(1,𝔽8)×Dic141, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3948, 2);