ΓL(1,𝔽₈)×#575 #13060736

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product	ΓL(1,𝔽₈)×#575
unambiguous semidirect product	(C₇×#575)⋊C₃

Splittings

Fitting, Hall	C₇×#575 ⋊ C₃
Sylow	C₇ ⋊ C₃×#575

Subgroups

	subgroup	G/H
center	C₄	C₂×D₄×ΓL(1,𝔽₈)
commutator, cosocle	C₄×C₇	C₂⁴×C₃
Frattini	C₄	C₂⁴×ΓL(1,𝔽₈)
Fitting	C₇×#575	C₃
socle	C₂×C₇	C₂²×C₃×D₄
hypercenter	#575	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#575
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #575

524 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 237 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 — 4 generators on 23 points:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#575, ΓL(1,𝔽₈)×#3585, #72229450, …
maximal normal subgroup of	AGL(1,𝔽₇)×#575, ΓL(1,𝔽₈)×#3585, #72229450, …
maximal quotient of	C₄×ΓL(1,𝔽₈)×AGL(1,ℤ/8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10025);