ΓL(1,𝔽₈)×#4184⋊C₃ #72249194

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

Aliases

direct product

ΓL(1,𝔽₈)×#4184⋊C₃

Splittings

Fitting	C₇×#6910 ⋊ C₂×C₃
derived	C₇×#731 ⋊ C₂×C₃²
Hall	ΓL(1,𝔽₈)×#6910 ⋊ C₂
Sylow	C₇ ⋊ C₃×#4184⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₇×#731	C₂×C₃²
Frattini	C₃²	C₃×C₃⋊D₃×ΓL(1,𝔽₈)
Fitting	C₇×#6910	C₂×C₃
socle	C₃×C₇	C₃²×#272
LCS	G ▷ C₇×#731
UCS	C₁

1084 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 1084 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)(10,16,13,25,18,15)(11,19,20,30,14,22)(17,34)(21,27)(23,31)(24,28,32,29,26,33);(2,3,4)(5,7,6)(8,20,19,12,14,22,9,11,30)(10,23,33,18,21,28,13,17,29)(15,26,27,16,32,31,25,24,34);(8,18,15,12,13,16,9,10,25)(11,17,26,20,23,32,14,21,24)(19,33,31,22,28,34,30,29,27)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#13091120, AGL(1,𝔽₇)×#4184⋊C₃
maximal normal subgroup of	AGL(1,𝔽₇)×#4184⋊C₃

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