ΓL(1,𝔽₈)×#568 #13060729

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

ΓL(1,𝔽₈)×#568

Splittings

Fitting, Hall	C₇×#568 ⋊ C₃
Sylow	C₇ ⋊ C₃×#568

Subgroups

	subgroup	G/H
center, Frattini	C₄	C₂⁴×ΓL(1,𝔽₈)
commutator, socle	C₂×C₇	C₂³×C₃×C₄
Fitting	C₇×#568	C₃
prime radical	C₂×#41×ΓL(1,𝔽₈)	C₂
cosocle	C₄×C₇	C₂⁴×C₃
hypercenter	#568	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#568
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ #568

516 conjugacy classes of subgroups; only the 357 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 357 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)(8,9,11,15,12,14,18,21)(10,13,17,20,16,19,22,23);(1,2,3,5,4,6,7)(8,10)(9,13)(11,17)(12,16)(14,19)(15,20)(18,22)(21,23);(10,16)(13,19)(17,22)(20,23);(9,14)(13,19)(15,21)(20,23)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#568, ΓL(1,𝔽₈)×#3585, #72229461, …
maximal normal subgroup of	AGL(1,𝔽₇)×#568, ΓL(1,𝔽₈)×#3585, #72229461, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10018);