AGL(1,𝔽₇)×C₈₉ #16836652

order	3738
isoclinism	isoclinic class of AGL(1,𝔽₇)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	AGL(1,𝔽₇)×C₈₉
unambiguous semidirect product	D₇⋊(C₃×C₈₉)
unambiguous semidirect product	ΓL(1,𝔽₈)⋊(C₂×C₈₉)
unambiguous semidirect product	(D₇×C₈₉)⋊C₃

Splittings

Fitting, socle, Hall	C₇×C₈₉ ⋊ C₂×C₃
derived, Sylow	C₇ ⋊ C₂×C₃×C₈₉
Hall	ΓL(1,𝔽₈)×C₈₉ ⋊ C₂
Hall	D₇×C₈₉ ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ C₂×C₈₉
Hall	D₇ ⋊ C₃×C₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₈₉	AGL(1,𝔽₇)
commutator, cosocle, hypocenter	C₇	C₂×C₃×C₈₉
Frattini	C₁	G
Fitting, socle	C₇×C₈₉	C₂×C₃
LCS	G ▷ C₇
UCS	C₁ ◁ C₈₉

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 623

A minimal faithful permutation representation — 2 generators on 96 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×AGL(1,𝔽₇)×C₈₉, C₈₉×C₃⋊AGL(1,𝔽₇), C₅×AGL(1,𝔽₇)×C₈₉, …
maximal normal subgroup of	C₃×AGL(1,𝔽₇)×C₈₉, C₅×AGL(1,𝔽₇)×C₈₉, C₇×AGL(1,𝔽₇)×C₈₉
maximal quotient of	C₈₉×D₇⋊C₉, C₃×AGL(1,𝔽₇)×C₈₉, C₈₉×C₃⋊AGL(1,𝔽₇), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3738, 11);

AGL(1,𝔽7)×C89 #16836652