ΓL(1,𝔽₈)×#455 #13054986

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

Aliases

direct product	ΓL(1,𝔽₈)×#455
unambiguous semidirect product	(C₇×#455)⋊C₃

Splittings

Fitting, Hall	C₇×#455 ⋊ C₃
Sylow	C₇ ⋊ C₃×#455

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×#455	C₃
socle	C₂×C₇	C₃×#120
prime radical	ΓL(1,𝔽₈)×H(5,𝔽₂)	C₂
hypercenter	#455	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#455
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #455

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

normal subgroups only

This group has 304 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)(10,16,18,23)(11,19,21,14)(13,20,22,17);(1,2,3,5,4,6,7)(8,10)(9,13)(11,20)(12,18)(14,23)(15,22)(16,19)(17,21);(11,21)(14,19)(16,23)(17,20)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#5933, ΓL(1,𝔽₈)×#3019, ΓL(1,𝔽₈)×#3288, …
maximal normal subgroup of	ΓL(1,𝔽₈)×#3019, ΓL(1,𝔽₈)×#3288, ΓL(1,𝔽₈)×#3018, …
maximal quotient of	ΓL(1,𝔽₈)×#1792

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 4275);