D₁₇₅₀₇ #46855875

order	35014
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₇₅₀₇
unambiguous semidirect product	C₇⋊D₂₅₀₁
unambiguous semidirect product	C₄₁⋊D₄₂₇
unambiguous semidirect product	C₆₁⋊D₂₈₇

Splittings

Fitting, socle, derived, Hall	C₇×C₄₁×C₆₁ ⋊ C₂
Hall	C₄₁×C₆₁ ⋊ D₇
Hall	C₇×C₆₁ ⋊ D₄₁
Hall	C₇×C₄₁ ⋊ D₆₁
Sylow	C₆₁ ⋊ D₂₈₇
Sylow	C₄₁ ⋊ D₄₂₇
Sylow	C₇ ⋊ D₂₅₀₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₄₁×C₆₁	C₂
LCS	G ▷ C₇×C₄₁×C₆₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 109
transitive degree	μ_t(G) = 17507

A minimal faithful permutation representation — 2 generators on 109 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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(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)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,80)

Polycyclic presentation:

2,7,41,61|b^a=b^6,c^a=c^40,d^a=d^60

Conjugacy classes & character table

8755 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35014, 8);

D17507 #46855875