ΓL(1,𝔽₈)×#2199 #72276916

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

ΓL(1,𝔽₈)×#2199

Splittings

Fitting, Hall	C₇×#2199 ⋊ C₃
Sylow	C₇ ⋊ C₃×#2199

Subgroups

	subgroup	G/H
center	C₂	ΓL(1,𝔽₈)×B(4,𝔽₂)
commutator, cosocle	C₂×C₇×D₄	C₂³×C₃
Frattini	C₂×D₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#2199	C₃
socle	C₂×C₇	C₃×B(4,𝔽₂)
hypercenter	#2199	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#2199
LCS	G ▷ C₂×C₇×D₄ ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #2199

540 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#2199
maximal normal subgroup of	AGL(1,𝔽₇)×#2199
maximal quotient of	ΓL(1,𝔽₈)×#13343

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