C₂×ΓL(1,𝔽₈)×C₁₀₃ #16864485

order	4326
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₁₀₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×ΓL(1,𝔽₈)×C₁₀₃
unambiguous semidirect product	(C₂×C₇)⋊(C₃×C₁₀₃)

Splittings

Fitting, socle, Hall	C₂×C₇×C₁₀₃ ⋊ C₃
derived, Sylow	C₇ ⋊ C₂×C₃×C₁₀₃
Hall	C₇×C₁₀₃ ⋊ C₂×C₃
Hall	C₂×C₇ ⋊ C₃×C₁₀₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₀₃	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₁₀₃
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₁₀₃	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₁₀₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 112
transitive degree	μ_t(G) = 1442

A minimal faithful permutation representation — 2 generators on 112 points:

(1,2)(4,5,6)(7,9,8)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112);(3,4,5,7,6,8,9)

Conjugacy classes & character table

1030 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×ΓL(1,𝔽₈)×C₁₀₃⋊C₃, D₃×ΓL(1,𝔽₈)×C₁₀₃, C₂×C₃×ΓL(1,𝔽₈)×C₁₀₃, …
maximal normal subgroup of	C₂×ΓL(1,𝔽₈)×C₁₀₃⋊C₃, C₂×C₃×ΓL(1,𝔽₈)×C₁₀₃, C₂×C₅×ΓL(1,𝔽₈)×C₁₀₃, …
maximal quotient of	C₂×C₇⋊C₉×C₁₀₃, D₃×ΓL(1,𝔽₈)×C₁₀₃, C₂×C₃×ΓL(1,𝔽₈)×C₁₀₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4326, 3);

C2×ΓL(1,𝔽8)×C103 #16864485