D₆₇₈₅ #28330841

order	13570
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₆₇₈₅
unambiguous semidirect product	C₅⋊D₁₃₅₇
unambiguous semidirect product	C₂₃⋊D₂₉₅
unambiguous semidirect product	C₅₉⋊D₁₁₅

Splittings

Fitting, socle, derived, Hall	C₅×C₂₃×C₅₉ ⋊ C₂
Hall	C₂₃×C₅₉ ⋊ D₅
Hall	C₅×C₅₉ ⋊ D₂₃
Hall	C₅×C₂₃ ⋊ D₅₉
Sylow	C₅₉ ⋊ D₁₁₅
Sylow	C₂₃ ⋊ D₂₉₅
Sylow	C₅ ⋊ D₁₃₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₂₃×C₅₉	C₂
LCS	G ▷ C₅×C₂₃×C₅₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 6785

A minimal faithful permutation representation — 2 generators on 87 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);(2,5)(3,4)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)

Polycyclic presentation:

2,5,23,59|b^a=b^4,c^a=c^22,d^a=d^58

Conjugacy classes & character table

3394 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅₉×D₁₁₅, D₂₃×D₂₉₅, D₅×D₁₃₅₇, …
maximal normal subgroup of	D₅₉×D₁₁₅, D₂₃×D₂₉₅, D₅×D₁₃₅₇, …
maximal quotient of	Dic₆₇₈₅, C₂×D₆₇₈₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13570, 8);

D6785 #28330841