ΓL(1,𝔽₈)×#456 #13054987

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

Aliases

direct product	ΓL(1,𝔽₈)×#456
unambiguous semidirect product	(C₇×#456)⋊C₃

Splittings

Fitting, Hall	C₇×#456 ⋊ C₃
Sylow	C₇ ⋊ C₃×#456

Subgroups

	subgroup	G/H
center	C₂	ΓL(1,𝔽₈)×#120
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#456	C₃
socle	C₂×C₇	C₃×#120
prime radical	ΓL(1,𝔽₈)×2₋¹⁺⁴	C₂
hypercenter	#456	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#456
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #456

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

normal subgroups only

This group has 284 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)(8,9,13,11)(10,16,19,23)(12,20,22,15)(14,21,17,18);(1,2,3,5,4,6,7)(8,10)(9,14)(11,17)(12,21)(13,19)(15,23)(16,20)(18,22);(8,11,13,9)(10,17,19,14)(12,15,22,20)(16,21,23,18)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#3291, ΓL(1,𝔽₈)×#3018
maximal normal subgroup of	ΓL(1,𝔽₈)×#3291, ΓL(1,𝔽₈)×#3018

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4276);