D₇₂₈₅ #28410092

order	14570
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₇₂₈₅
unambiguous semidirect product	C₅⋊D₁₄₅₇
unambiguous semidirect product	C₃₁⋊D₂₃₅
unambiguous semidirect product	C₄₇⋊D₁₅₅

Splittings

Fitting, socle, derived, Hall	C₅×C₃₁×C₄₇ ⋊ C₂
Hall	C₃₁×C₄₇ ⋊ D₅
Hall	C₅×C₄₇ ⋊ D₃₁
Hall	C₅×C₃₁ ⋊ D₄₇
Sylow	C₄₇ ⋊ D₁₅₅
Sylow	C₃₁ ⋊ D₂₃₅
Sylow	C₅ ⋊ D₁₄₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₃₁×C₄₇	C₂
LCS	G ▷ C₅×C₃₁×C₄₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 7285

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)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)

Polycyclic presentation:

2,5,31,47|b^a=b^4,c^a=c^30,d^a=d^46

Conjugacy classes & character table

3644 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₇×D₁₅₅, D₃₁×D₂₃₅, D₅×D₁₄₅₇, …
maximal normal subgroup of	D₄₇×D₁₅₅, D₃₁×D₂₃₅, D₅×D₁₄₅₇, …
maximal quotient of	Dic₇₂₈₅, C₂×D₇₂₈₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14570, 9);

D7285 #28410092