ΓL(1,𝔽₈)×#4183⋊C₃ #72249182

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

Aliases

direct product

ΓL(1,𝔽₈)×#4183⋊C₃

Splittings

Fitting	C₇×#6909 ⋊ C₂×C₃
derived	C₇×AΣL(1,𝔽₂₇) ⋊ C₂×C₃²
Hall	ΓL(1,𝔽₈)×#6909 ⋊ C₂
Sylow	C₇ ⋊ C₃×#4183⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₇×AΣL(1,𝔽₂₇)	C₂×C₃²
Frattini	C₃²	C₃×C₃⋊D₃×ΓL(1,𝔽₈)
Fitting	C₇×#6909	C₂×C₃
socle	C₃×C₇	C₃²×#272
LCS	G ▷ C₇×AΣL(1,𝔽₂₇)
UCS	C₁

1156 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 1156 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)(9,21,16,13,14,23)(10,28,20,17,18,30)(11,12)(15,33,19,24,25,31)(22,27)(26,34)(29,32);(2,3,4)(5,7,6)(8,10,17)(9,15,27)(11,20,28)(12,18,30)(13,22,24)(14,19,29)(16,25,34)(21,32,31)(23,26,33);(8,9,13)(10,19,32)(11,16,21)(12,14,23)(15,26,20)(17,29,31)(18,25,22)(24,30,34)(27,33,28)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(1,𝔽₈)×#13091120, AGL(1,𝔽₇)×#4183⋊C₃
maximal normal subgroup of	AGL(1,𝔽₇)×#4183⋊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)(9,21,16,13,14,23)(10,28,20,17,18,30)(11,12)(15,33,19,24,25,31)(22,27)(26,34)(29,32), (2,3,4)(5,7,6)(8,10,17)(9,15,27)(11,20,28)(12,18,30)(13,22,24)(14,19,29)(16,25,34)(21,32,31)(23,26,33), (8,9,13)(10,19,32)(11,16,21)(12,14,23)(15,26,20)(17,29,31)(18,25,22)(24,30,34)(27,33,28));