C₂×ΓL(1,𝔽₈)×C₄₇ #14520320

order	1974
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×ΓL(1,𝔽₈)×C₄₇
unambiguous semidirect product	(C₂×C₇)⋊(C₃×C₄₇)
unambiguous semidirect product	(C₂×C₇×C₄₇)⋊C₃

Splittings

Fitting, socle, Hall	C₂×C₇×C₄₇ ⋊ C₃
derived, Sylow	C₇ ⋊ C₂×C₃×C₄₇
Hall	C₇×C₄₇ ⋊ C₂×C₃
Hall	C₂×C₇ ⋊ C₃×C₄₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄₇	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₄₇
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₄₇	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₄₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 658

A minimal faithful permutation representation — 2 generators on 56 points:

(1,2)(4,5,6)(7,9,8)(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);(3,4,5,7,6,8,9)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×Dic₄₇, C₄₇×ΓL(1,𝔽₈)⋊C₄, ΓL(1,𝔽₈)⋊Dic₄₇, …
maximal normal subgroup of	ΓL(1,𝔽₈)×Dic₄₇, C₄₇×ΓL(1,𝔽₈)⋊C₄, ΓL(1,𝔽₈)⋊Dic₄₇, …
maximal quotient of	C₄×ΓL(1,𝔽₈)×C₄₇, C₂²×ΓL(1,𝔽₈)×C₄₇, C₂×C₄₇×C₄₉⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1974, 4);

C2×ΓL(1,𝔽8)×C47 #14520320