ΓL(1,𝔽₈)×D₃₀₁ #27162441

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

Aliases

direct product

ΓL(1,𝔽₈)×D₃₀₁

Splittings

Fitting, socle, derived, Hall	C₇²×C₄₃ ⋊ C₂×C₃
Hall	C₇×ΓL(1,𝔽₈)×C₄₃ ⋊ C₂
Hall	C₇×D₃₀₁ ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ D₄₃
Sylow	C₇² ⋊ C₃×D₄₃
Sylow	C₄₃ ⋊ D₇×ΓL(1,𝔽₈)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₄₃	C₂×C₃
LCS	G ▷ C₇²×C₄₃
UCS	C₁

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	AGL(1,𝔽₇)×D₃₀₁, (ΓL(1,𝔽₈)⋊D₇)⋊D₄₃, D₇×ΓL(1,𝔽₈)×D₄₃, …
maximal normal subgroup of	AGL(1,𝔽₇)×D₃₀₁, (ΓL(1,𝔽₈)⋊D₇)⋊D₄₃, D₇×ΓL(1,𝔽₈)×D₄₃, …
maximal quotient of	ΓL(1,𝔽₈)×Dic₃₀₁, C₂×ΓL(1,𝔽₈)×D₃₀₁, C₇⋊C₉×D₃₀₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12642, 106);

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