ΓL(1,𝔽₈)×C₆₁ #13046194

order	1281
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₃×C₆₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	ΓL(1,𝔽₈)×C₆₁
unambiguous semidirect product	C₇⋊(C₃×C₆₁)

Splittings

Fitting, socle, Hall	C₇×C₆₁ ⋊ C₃
derived, Sylow	C₇ ⋊ C₃×C₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₆₁	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₃×C₆₁
Frattini	C₁	G
Fitting, socle	C₇×C₆₁	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₆₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 427

A minimal faithful permutation representation — 2 generators on 68 points:

(2,3,4)(5,7,6)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68);(1,2,3,5,4,6,7)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×ΓL(1,𝔽₈)×C₆₁, ΓL(1,𝔽₈)×D₆₁, AGL(1,𝔽₇)×C₆₁, …
maximal normal subgroup of	C₂×ΓL(1,𝔽₈)×C₆₁, ΓL(1,𝔽₈)×D₆₁, AGL(1,𝔽₇)×C₆₁, …
maximal quotient of	C₂×ΓL(1,𝔽₈)×C₆₁, C₆₁×C₇⋊C₉, C₃×ΓL(1,𝔽₈)×C₆₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1281, 1);

ΓL(1,𝔽8)×C61 #13046194