ΓL(1,𝔽₈)×#458 #13054989

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

ΓL(1,𝔽₈)×#458

Splittings

Fitting, Hall	C₇×#458 ⋊ C₃
Sylow	C₇ ⋊ C₃×#458

Subgroups

	subgroup	G/H
center	C₂	ΓL(1,𝔽₈)×#120
commutator, cosocle	C₂³×C₇	C₂³×C₃
Frattini	C₂³	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#458	C₃
socle	C₂×C₇	C₃×#120
prime radical	ΓL(1,𝔽₈)×H(5,𝔽₂)	C₂
hypercenter	#458	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#458
LCS	G ▷ C₂³×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #458

320 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 81 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)(8,9,12,15)(10,13,17,20)(11,14,18,21)(16,22,23,19);(1,2,3,5,4,6,7)(8,10)(9,13)(11,16)(12,17)(14,19)(15,20)(18,23)(21,22);(8,11)(9,14)(10,16)(12,18)(13,19)(15,21)(17,23)(20,22)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277002, AGL(1,𝔽₇)×#458, ΓL(1,𝔽₈)×#3026, …
maximal normal subgroup of	#72277002, AGL(1,𝔽₇)×#458, ΓL(1,𝔽₈)×#3026, …
maximal quotient of	ΓL(1,𝔽₈)×#1790

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4278);