ΓL(1,𝔽₈)×#92300 #72249179

order	10206
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃³

Aliases

direct product

ΓL(1,𝔽₈)×#92300

Splittings

Fitting	C₇×C₃²⋊H(3,𝔽₃) ⋊ C₂×C₃
derived	C₃×C₇×H(3,𝔽₃) ⋊ C₂×C₃²
Hall	ΓL(1,𝔽₈)×C₃²⋊H(3,𝔽₃) ⋊ C₂
Sylow	C₇ ⋊ C₃×#92300

Subgroups

	subgroup	G/H
center, hypercenter	C₃	ΓL(1,𝔽₈)×C₃⋊((C₃⋊D₃)⋊C₃)
commutator, cosocle, hypocenter	C₃×C₇×H(3,𝔽₃)	C₂×C₃²
Frattini	C₃²	C₃×C₃⋊D₃×ΓL(1,𝔽₈)
Fitting	C₇×C₃²⋊H(3,𝔽₃)	C₂×C₃
socle	C₃²×C₇	C₃²×C₃⋊D₃
LCS	G ▷ C₃×C₇×H(3,𝔽₃)
UCS	C₁ ◁ C₃

2758 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 2758 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:

(2,7,9,4,5,8)(3,6)(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);(1,2,4)(3,5,8)(6,9,7)(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)×#92300
maximal normal subgroup of	AGL(1,𝔽₇)×#92300

GAP

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

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