D₁₅₅₅₅ #44527446

order	31110
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₅₅₅
unambiguous semidirect product	C₃⋊D₅₁₈₅
unambiguous semidirect product	C₅⋊D₃₁₁₁
unambiguous semidirect product	C₁₇⋊D₉₁₅
unambiguous semidirect product	C₆₁⋊D₂₅₅

Splittings

Fitting, socle, derived, Hall	C₃×C₅×C₁₇×C₆₁ ⋊ C₂
Hall	C₅×C₁₇×C₆₁ ⋊ D₃
Hall	C₃×C₁₇×C₆₁ ⋊ D₅
Hall	C₁₇×C₆₁ ⋊ D₁₅
Hall	C₃×C₅×C₆₁ ⋊ D₁₇
Hall	C₅×C₆₁ ⋊ D₅₁
Hall	C₃×C₅×C₁₇ ⋊ D₆₁
Hall	C₃×C₆₁ ⋊ D₈₅
Hall	C₅×C₁₇ ⋊ D₁₈₃
Sylow	C₆₁ ⋊ D₂₅₅
Hall	C₃×C₁₇ ⋊ D₃₀₅
Sylow	C₁₇ ⋊ D₉₁₅
Hall	C₃×C₅ ⋊ D₁₀₃₇
Sylow	C₅ ⋊ D₃₁₁₁
Sylow	C₃ ⋊ D₅₁₈₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₁₇×C₆₁	C₂
LCS	G ▷ C₃×C₅×C₁₇×C₆₁
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 86
transitive degree	μ_t(G) = 15555

A minimal faithful permutation representation — 2 generators on 86 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);(2,3)(5,8)(6,7)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2,3,5,17,61|b^a=b^2,c^a=c^4,d^a=d^16,e^a=e^60

Conjugacy classes & character table

7779 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₅₅₅
maximal normal subgroup of	C₂×D₁₅₅₅₅
maximal quotient of	Dic₁₅₅₅₅, C₂×D₁₅₅₅₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31110, 19);

D15555 #44527446