ΓL(1,𝔽₈)×#92333 #72249190

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

Aliases

direct product

ΓL(1,𝔽₈)×#92333

Splittings

Fitting	C₇×#6911 ⋊ C₂×C₃
derived	C₇×#732 ⋊ C₂×C₃²
Hall	ΓL(1,𝔽₈)×#6911 ⋊ C₂
Sylow	C₇ ⋊ C₃×#92333

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₇×#732	C₂×C₃²
Frattini	C₃²	C₃×C₃⋊D₃×ΓL(1,𝔽₈)
Fitting	C₇×#6911	C₂×C₃
socle	C₃×C₇	C₃²×#272
LCS	G ▷ C₇×#732
UCS	C₁

1228 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 1228 conjugacy classes of subgroups — too many to draw. Only its 85 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) = 34
transitive degree	μ_t(G) = 189

A minimal faithful permutation representation — 3 generators on 34 points:

(1,2,3,5,4,6,7)(8,9)(10,15,16,17,13,22)(11,19,20,25,14,23)(18,32,21,33,24,28)(26,31)(27,30)(29,34);(2,3,4)(5,7,6)(8,11,19)(9,14,25)(10,18,30)(12,20,23)(13,21,31)(15,26,28)(16,24,34)(17,27,33)(22,29,32);(8,13,15,9,16,17,12,10,22)(11,24,29,14,18,26,20,21,27)(19,34,32,25,30,28,23,31,33)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#13091120, AGL(1,𝔽₇)×#92333
maximal normal subgroup of	AGL(1,𝔽₇)×#92333

GAP

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

gap> G := Group((1,2,3,5,4,6,7)(8,9)(10,15,16,17,13,22)(11,19,20,25,14,23)(18,32,21,33,24,28)(26,31)(27,30)(29,34), (2,3,4)(5,7,6)(8,11,19)(9,14,25)(10,18,30)(12,20,23)(13,21,31)(15,26,28)(16,24,34)(17,27,33)(22,29,32), (8,13,15,9,16,17,12,10,22)(11,24,29,14,18,26,20,21,27)(19,34,32,25,30,28,23,31,33));