ΓL(1,𝔽₈)×#2956 #72229621

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

ΓL(1,𝔽₈)×#2956

Splittings

Fitting, Hall	C₇×#2956 ⋊ C₃
Sylow	C₇ ⋊ C₃×#2956

Subgroups

	subgroup	G/H
center	C₄	C₂²×D₄×ΓL(1,𝔽₈)
commutator	C₄×C₇	C₂³×C₃×C₄
Frattini	C₂×C₄	C₂⁴×ΓL(1,𝔽₈)
Fitting	C₇×#2956	C₃
socle	C₂×C₇	C₂×C₃×C₄×D₄
prime radical	ΓL(1,𝔽₈)×#575	C₂
cosocle	C₂×C₄×C₇	C₂⁴×C₃
hypercenter	#2956	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#2956
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #2956

916 conjugacy classes of subgroups.

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

Conjugacy classes & character table

220 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)(8,9,12,16)(10,17,18,21)(11,13,19,14)(15,22,23,20), (1,2,4,6,3,5,7)(9,13,16,14)(10,18)(15,17,23,21)(20,22), (9,14,16,13)(15,17,23,21), (8,10,19,22,12,18,11,20)(9,15,13,21,16,23,14,17));

ΓL(1,𝔽8)×#2956 #72229621