D₇₈₈₁ #30628798

order	15762
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₇₈₈₁
unambiguous semidirect product	C₃⋊D₂₆₂₇
unambiguous semidirect product	C₃₇⋊D₂₁₃
unambiguous semidirect product	C₇₁⋊D₁₁₁

Splittings

Fitting, socle, derived, Hall	C₃×C₃₇×C₇₁ ⋊ C₂
Hall	C₃₇×C₇₁ ⋊ D₃
Hall	C₃×C₇₁ ⋊ D₃₇
Hall	C₃×C₃₇ ⋊ D₇₁
Sylow	C₇₁ ⋊ D₁₁₁
Sylow	C₃₇ ⋊ D₂₁₃
Sylow	C₃ ⋊ D₂₆₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₃₇×C₇₁	C₂
LCS	G ▷ C₃×C₃₇×C₇₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 7881

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);(2,3)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(42,111)(43,110)(44,109)(45,108)(46,107)(47,106)(48,105)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)

Polycyclic presentation:

2,3,37,71|b^a=b^2,c^a=c^36,d^a=d^70

Conjugacy classes & character table

3942 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₇₁×D₁₁₁, D₃₇×D₂₁₃, D₃×D₂₆₂₇, …
maximal normal subgroup of	D₇₁×D₁₁₁, D₃₇×D₂₁₃, D₃×D₂₆₂₇, …
maximal quotient of	Dic₇₈₈₁, C₂×D₇₈₈₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15762, 9);

D7881 #30628798