D₇₅₈₅ #29503196

order	15170
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₇₅₈₅
unambiguous semidirect product	C₅⋊D₁₅₁₇
unambiguous semidirect product	C₃₇⋊D₂₀₅
unambiguous semidirect product	C₄₁⋊D₁₈₅

Splittings

Fitting, socle, derived, Hall	C₅×C₃₇×C₄₁ ⋊ C₂
Hall	C₃₇×C₄₁ ⋊ D₅
Hall	C₅×C₄₁ ⋊ D₃₇
Hall	C₅×C₃₇ ⋊ D₄₁
Sylow	C₄₁ ⋊ D₁₈₅
Sylow	C₃₇ ⋊ D₂₀₅
Sylow	C₅ ⋊ D₁₅₁₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₃₇×C₄₁	C₂
LCS	G ▷ C₅×C₃₇×C₄₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 7585

A minimal faithful permutation representation — 2 generators on 83 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);(2,5)(3,4)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)

Polycyclic presentation:

2,5,37,41|b^a=b^4,c^a=c^36,d^a=d^40

Conjugacy classes & character table

3794 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#44476845, #44476846, #44476849, …
maximal normal subgroup of	#44476845, #44476846, #44476849, …
maximal quotient of	Dic₇₅₈₅, C₂×D₇₅₈₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15170, 9);

D7585 #29503196