D₁₅₈₄₁ #44540027

order	31682
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₈₄₁
unambiguous semidirect product	C₇⋊D₂₂₆₃
unambiguous semidirect product	C₃₁⋊D₅₁₁
unambiguous semidirect product	C₇₃⋊D₂₁₇

Splittings

Fitting, socle, derived, Hall	C₇×C₃₁×C₇₃ ⋊ C₂
Hall	C₃₁×C₇₃ ⋊ D₇
Hall	C₇×C₇₃ ⋊ D₃₁
Hall	C₇×C₃₁ ⋊ D₇₃
Sylow	C₇₃ ⋊ D₂₁₇
Sylow	C₃₁ ⋊ D₅₁₁
Sylow	C₇ ⋊ D₂₂₆₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₃₁×C₇₃	C₂
LCS	G ▷ C₇×C₃₁×C₇₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 15841

A minimal faithful permutation representation — 2 generators on 111 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)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(2,7)(3,6)(4,5)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Polycyclic presentation:

2,7,31,73|b^a=b^6,c^a=c^30,d^a=d^72

Conjugacy classes & character table

7922 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₈₄₁
maximal normal subgroup of	C₂×D₁₅₈₄₁
maximal quotient of	Dic₁₅₈₄₁, C₂×D₁₅₈₄₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31682, 8);

D15841 #44540027