C₂×ΓL(1,𝔽₈)×#131 #13060728

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

Aliases

direct product

C₂×ΓL(1,𝔽₈)×#131

Splittings

Fitting, Hall	C₂×C₇×#131 ⋊ C₃
Sylow	C₇ ⋊ C₂×C₃×#131

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×ΓL(1,𝔽₈)
commutator	C₂×C₇	C₂³×C₃×C₄
Frattini	C₄	C₂⁴×ΓL(1,𝔽₈)
Fitting	C₂×C₇×#131	C₃
socle	C₂²×C₇	C₂²×C₃×C₄
prime radical	C₂×#41×ΓL(1,𝔽₈)	C₂
cosocle	C₄×C₇	C₂⁴×C₃
hypercenter	C₂×#131	ΓL(1,𝔽₈)
hypocenter	C₇	C₂×C₃×#131
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

532 conjugacy classes of subgroups; only the 363 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 363 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) = 25
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 25 points:

(1,2)(20,21,22)(23,25,24);(3,4,6,9,7,10,13,16)(5,8,12,15,11,14,17,18)(19,20,21,23,22,24,25);(3,5)(4,8)(6,12)(7,11)(9,15)(10,14)(13,17)(16,18);(5,11)(8,14)(12,17)(15,18)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10017);