C₂×ΓL(1,𝔽₈)×C₈₉ #16836643

order	3738
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×ΓL(1,𝔽₈)×C₈₉
unambiguous semidirect product	(C₂×C₇)⋊(C₃×C₈₉)
unambiguous semidirect product	(C₂×C₇×C₈₉)⋊C₃

Splittings

Fitting, socle, Hall	C₂×C₇×C₈₉ ⋊ C₃
derived, Sylow	C₇ ⋊ C₂×C₃×C₈₉
Hall	C₇×C₈₉ ⋊ C₂×C₃
Hall	C₂×C₇ ⋊ C₃×C₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₈₉	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₈₉
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₈₉	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₈₉

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 98
transitive degree	μ_t(G) = 1246

A minimal faithful permutation representation — 2 generators on 98 points:

(1,2)(4,5,6)(7,9,8)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98);(3,4,5,7,6,8,9)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₃×ΓL(1,𝔽₈)×C₈₉, C₂×C₃×ΓL(1,𝔽₈)×C₈₉, C₂×C₅×ΓL(1,𝔽₈)×C₈₉, …
maximal normal subgroup of	C₂×C₃×ΓL(1,𝔽₈)×C₈₉, C₂×C₅×ΓL(1,𝔽₈)×C₈₉, C₂×C₇×ΓL(1,𝔽₈)×C₈₉
maximal quotient of	C₂×C₇⋊C₉×C₈₉, D₃×ΓL(1,𝔽₈)×C₈₉, C₂×C₃×ΓL(1,𝔽₈)×C₈₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3738, 2);

C2×ΓL(1,𝔽8)×C89 #16836643