D₇×ΓL(1,𝔽₈)×C₄₃ #27162436

order	12642
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄₃ · Schur multiplier (H₂) = C₁

Aliases

direct product

D₇×ΓL(1,𝔽₈)×C₄₃

Splittings

Fitting, socle, Hall	C₇²×C₄₃ ⋊ C₂×C₃
derived, Sylow	C₇² ⋊ C₂×C₃×C₄₃
Hall	C₇×ΓL(1,𝔽₈)×C₄₃ ⋊ C₂
Hall	C₇×D₇×C₄₃ ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ C₂×C₄₃
Hall	C₇×D₇ ⋊ C₃×C₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₄₃	D₇×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇²	C₂×C₃×C₄₃
Frattini	C₁	G
Fitting, socle	C₇²×C₄₃	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₄₃

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 1806

A minimal faithful permutation representation — 2 generators on 57 points:

(2,3)(4,5)(6,7)(9,10,11)(12,14,13)(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);(1,2,4,6,7,5,3)(8,9,10,12,11,13,14)

Conjugacy classes & character table

1075 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇×AGL(1,𝔽₇)×C₄₃, D₇×ΓL(1,𝔽₈)⋊D₄₃, D₇×ΓL(1,𝔽₈)×D₄₃, …
maximal normal subgroup of	D₇×AGL(1,𝔽₇)×C₄₃, D₇×ΓL(1,𝔽₈)⋊D₄₃, D₇×ΓL(1,𝔽₈)×D₄₃, …
maximal quotient of	ΓL(1,𝔽₈)×Dic₇×C₄₃, C₂×D₇×ΓL(1,𝔽₈)×C₄₃, D₇×C₄₃×C₇⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12642, 101);

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