ΓL(1,𝔽₈)×#1906 #72229528

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

ΓL(1,𝔽₈)×#1906

Splittings

Fitting, Hall	C₇×#1906 ⋊ C₃
Sylow	C₇ ⋊ C₃×#1906

Subgroups

	subgroup	G/H
center	C₄	ΓL(1,𝔽₈)×#121
commutator	C₂×C₄×C₇	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#1906	C₃
socle	C₂×C₇	C₃×#386
prime radical	ΓL(1,𝔽₈)×#575	C₂
cosocle	C₂²×C₄×C₇	C₂³×C₃
hypercenter	#1906	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#1906
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1906

664 conjugacy classes of subgroups.

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

ΓL(1,𝔽8)×#1906 #72229528