D₁₅₆₉₁ #44535971

order	31382
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₆₉₁
unambiguous semidirect product	C₁₃⋊D₁₂₀₇
unambiguous semidirect product	C₁₇⋊D₉₂₃
unambiguous semidirect product	C₇₁⋊D₂₂₁

Splittings

Fitting, socle, derived, Hall	C₁₃×C₁₇×C₇₁ ⋊ C₂
Hall	C₁₇×C₇₁ ⋊ D₁₃
Hall	C₁₃×C₇₁ ⋊ D₁₇
Hall	C₁₃×C₁₇ ⋊ D₇₁
Sylow	C₇₁ ⋊ D₂₂₁
Sylow	C₁₇ ⋊ D₉₂₃
Sylow	C₁₃ ⋊ D₁₂₀₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₃×C₁₇×C₇₁	C₂
LCS	G ▷ C₁₃×C₁₇×C₇₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 15691

A minimal faithful permutation representation — 2 generators on 101 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);(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(32,101)(33,100)(34,99)(35,98)(36,97)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)

Polycyclic presentation:

2,13,17,71|b^a=b^12,c^a=c^16,d^a=d^70

Conjugacy classes & character table

7847 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₆₉₁
maximal normal subgroup of	C₂×D₁₅₆₉₁
maximal quotient of	Dic₁₅₆₉₁, C₂×D₁₅₆₉₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31382, 8);

D15691 #44535971