C₂×ΓL(1,𝔽₈)×C₇₉ #15693286

order	3318
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₇₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂×ΓL(1,𝔽₈)×C₇₉
unambiguous semidirect product	(C₂×C₇)⋊(C₃×C₇₉)

Splittings

Fitting, socle, Hall	C₂×C₇×C₇₉ ⋊ C₃
derived, Sylow	C₇ ⋊ C₂×C₃×C₇₉
Hall	C₇×C₇₉ ⋊ C₂×C₃
Hall	C₂×C₇ ⋊ C₃×C₇₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₇₉	ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₇₉
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₇₉	C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₇₉

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 88
transitive degree	μ_t(G) = 1106

A minimal faithful permutation representation — 2 generators on 88 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);(3,4,5,7,6,8,9)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₅×ΓL(1,𝔽₈)×C₇₉, D₅×ΓL(1,𝔽₈)×C₇₉, C₂×C₇₉×C₄₉⋊C₃, …
maximal normal subgroup of	C₂×C₅×ΓL(1,𝔽₈)×C₇₉, C₂×C₇×ΓL(1,𝔽₈)×C₇₉, C₂×C₁₁×ΓL(1,𝔽₈)×C₇₉
maximal quotient of	C₂×C₅×ΓL(1,𝔽₈)×C₇₉, D₅×ΓL(1,𝔽₈)×C₇₉, C₂×C₇₉×C₄₉⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3318, 3);

C2×ΓL(1,𝔽8)×C79 #15693286