D₁₂₃₄₁ #37590550

order	24682
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₃₄₁
unambiguous semidirect product	C₇⋊D₁₇₆₃
unambiguous semidirect product	C₄₁⋊D₃₀₁
unambiguous semidirect product	C₄₃⋊D₂₈₇

Splittings

Fitting, socle, derived, Hall	C₇×C₄₁×C₄₃ ⋊ C₂
Hall	C₄₁×C₄₃ ⋊ D₇
Hall	C₇×C₄₃ ⋊ D₄₁
Hall	C₇×C₄₁ ⋊ D₄₃
Sylow	C₄₃ ⋊ D₂₈₇
Sylow	C₄₁ ⋊ D₃₀₁
Sylow	C₇ ⋊ D₁₇₆₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₄₁×C₄₃	C₂
LCS	G ▷ C₇×C₄₁×C₄₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 12341

A minimal faithful permutation representation — 2 generators on 91 points:

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

Polycyclic presentation:

2,7,41,43|b^a=b^6,c^a=c^40,d^a=d^42

Conjugacy classes & character table

6172 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24682, 9);

D12341 #37590550