ΓL(1,𝔽₈)×#92302 #72249186

order	10206
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product

ΓL(1,𝔽₈)×#92302

Splittings

Fitting	C₇×#6891 ⋊ C₂×C₃
derived	C₇×C₃²⋊C₉ ⋊ C₂×C₃²
Hall	ΓL(1,𝔽₈)×#6891 ⋊ C₂
Sylow	C₇ ⋊ C₃×#92302

Subgroups

	subgroup	G/H
center, hypercenter	C₃	ΓL(1,𝔽₈)×C₃⋊AGL(1,ℤ/9)
commutator, cosocle, hypocenter	C₇×C₃²⋊C₉	C₂×C₃²
Frattini	C₃²	C₃×C₃⋊D₃×ΓL(1,𝔽₈)
Fitting	C₇×#6891	C₂×C₃
socle	C₃²×C₇	C₃²×C₃⋊D₃
LCS	G ▷ C₇×C₃²⋊C₉
UCS	C₁ ◁ C₃

2092 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 2092 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 126

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#13091120, AGL(1,𝔽₇)×#92302
maximal normal subgroup of	AGL(1,𝔽₇)×#92302

GAP

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

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