ΓL(1,𝔽₈)×#3490 #72276997

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

ΓL(1,𝔽₈)×#3490

Splittings

Fitting, Hall	C₇×#3490 ⋊ C₃
Sylow	C₇ ⋊ C₃×#3490

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂²×C₇	C₂⁵×C₃
Fitting	C₇×#3490	C₃
hypercenter	#3490	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#3490
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #3490

2656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 5 generators on 23 points:

(2,3,4)(5,7,6)(10,15)(12,14);(1,2,3,5,4,6,7)(17,21)(20,23);(18,22)(20,23);(8,10)(9,12)(11,15)(13,14)(16,18)(17,20)(19,22)(21,23);(8,9,11,13)(10,14,15,12)(16,17)(18,20)(19,21)(22,23)

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#62817, AGL(1,𝔽₇)×#3490, #72277873
maximal normal subgroup of	ΓL(1,𝔽₈)×#62817, AGL(1,𝔽₇)×#3490, #72277873

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,3,4)(5,7,6)(10,15)(12,14), (1,2,3,5,4,6,7)(17,21)(20,23), (18,22)(20,23), (8,10)(9,12)(11,15)(13,14)(16,18)(17,20)(19,22)(21,23), (8,9,11,13)(10,14,15,12)(16,17)(18,20)(19,21)(22,23));

ΓL(1,𝔽8)×#3490 #72276997