ΓL(1,𝔽₈)×#3026 #72277096

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

ΓL(1,𝔽₈)×#3026

Splittings

Fitting, Hall	C₇×#3026 ⋊ C₃
Sylow	C₇ ⋊ C₃×#3026

Subgroups

	subgroup	G/H
center	C₂	C₂×ΓL(1,𝔽₈)×#120
commutator, cosocle	C₂³×C₇	C₂⁴×C₃
Frattini	C₂³	C₂⁴×ΓL(1,𝔽₈)
Fitting	C₇×#3026	C₃
socle	C₂×C₇	C₂×C₃×#120
hypercenter	#3026	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#3026
LCS	G ▷ C₂³×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #3026

1600 conjugacy classes of subgroups; only the 318 normal subgroups are shown.

normal subgroups only

This group has 1600 conjugacy classes of subgroups — too many to draw. Only its 318 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)(10,17)(13,20)(14,21)(16,22);(1,2,4,6,3,5,7)(8,11)(9,14)(10,16)(12,18)(13,19)(15,21)(17,22)(20,23);(8,10)(9,13)(11,16)(12,17)(14,19)(15,20)(18,22)(21,23);(8,9)(10,13)(11,14)(12,15)(16,19)(17,20)(18,21)(22,23)

Conjugacy classes & character table

145 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#3026
maximal normal subgroup of	AGL(1,𝔽₇)×#3026
maximal quotient of	ΓL(1,𝔽₈)×#32898

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