C₁₁×C₂₉×C₇⋊C₉ #34117136

order	20097
isoclinism	isoclinic class of ΓL(1,𝔽₈)
automorphism	?
homology	Abelianization (H₁) = C₉×C₁₁×C₂₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₁×C₂₉×C₇⋊C₉
unambiguous semidirect product	C₇⋊(C₉×C₁₁×C₂₉)
unambiguous semidirect product	(C₇×C₁₁)⋊(C₉×C₂₉)
unambiguous semidirect product	(C₇×C₂₉)⋊(C₉×C₁₁)
unambiguous semidirect product	(C₇×C₁₁×C₂₉)⋊C₉

Splittings

derived, Sylow	C₇ ⋊ C₉×C₁₁×C₂₉
Hall	C₇×C₁₁×C₂₉ ⋊ C₉
Hall	C₇×C₂₉ ⋊ C₉×C₁₁
Hall	C₇×C₁₁ ⋊ C₉×C₂₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₁₁×C₂₉	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₉×C₁₁×C₂₉
Frattini	C₃	C₁₁×ΓL(1,𝔽₈)×C₂₉
Fitting, socle, prime radical	C₃×C₇×C₁₁×C₂₉	C₃
cosocle	C₃×C₇	C₃×C₁₁×C₂₉
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₁₁×C₂₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 20097

A minimal faithful permutation representation — 2 generators on 56 points:

(2,3,4)(5,7,6)(8,15,14,10,9,16,13,12,11)(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);(1,2,3,5,4,6,7)

Conjugacy classes & character table

4785 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20097, 3);

C11×C29×C7⋊C9 #34117136