D₇₈₈₅ #30628827

order	15770
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₇₈₈₅
unambiguous semidirect product	C₅⋊D₁₅₇₇
unambiguous semidirect product	C₁₉⋊D₄₁₅
unambiguous semidirect product	C₈₃⋊D₉₅

Splittings

Fitting, socle, derived, Hall	C₅×C₁₉×C₈₃ ⋊ C₂
Hall	C₁₉×C₈₃ ⋊ D₅
Hall	C₅×C₈₃ ⋊ D₁₉
Hall	C₅×C₁₉ ⋊ D₈₃
Sylow	C₈₃ ⋊ D₉₅
Sylow	C₁₉ ⋊ D₄₁₅
Sylow	C₅ ⋊ D₁₅₇₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₁₉×C₈₃	C₂
LCS	G ▷ C₅×C₁₉×C₈₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 7885

A minimal faithful permutation representation — 2 generators on 107 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);(2,5)(3,4)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(26,107)(27,106)(28,105)(29,104)(30,103)(31,102)(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:

2,5,19,83|b^a=b^4,c^a=c^18,d^a=d^82

Conjugacy classes & character table

3944 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₈₃×D₉₅, D₁₉×D₄₁₅, D₅×D₁₅₇₇, …
maximal normal subgroup of	D₈₃×D₉₅, D₁₉×D₄₁₅, D₅×D₁₅₇₇, …
maximal quotient of	Dic₇₈₈₅, C₂×D₇₈₈₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15770, 8);

D7885 #30628827