C₄₃×ΓL(1,𝔽₈)⋊D₇ #27162425

order	12642
isoclinism	isoclinic class of ΓL(1,𝔽₈)⋊D₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₄₃×ΓL(1,𝔽₈)⋊D₇
unambiguous semidirect product	ΓL(1,𝔽₈)⋊(D₇×C₄₃)

Splittings

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

Subgroups

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

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 903

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

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

Conjugacy classes & character table

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

Relationships

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12642, 90);

C43×ΓL(1,𝔽8)⋊D7 #27162425