D₁₁₆₅₅ #37552001

order	23310
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₆₅₅
unambiguous semidirect product	C₅⋊D₂₃₃₁
unambiguous semidirect product	C₇⋊D₁₆₆₅
unambiguous semidirect product	C₉⋊D₁₂₉₅
unambiguous semidirect product	C₃₇⋊D₃₁₅

Splittings

Fitting, derived, Hall	C₅×C₇×C₉×C₃₇ ⋊ C₂
Hall	C₇×C₉×C₃₇ ⋊ D₅
Hall	C₅×C₉×C₃₇ ⋊ D₇
Hall	C₅×C₇×C₃₇ ⋊ D₉
Hall	C₉×C₃₇ ⋊ D₃₅
Hall	C₅×C₇×C₉ ⋊ D₃₇
Hall	C₇×C₃₇ ⋊ D₄₅
Hall	C₅×C₃₇ ⋊ D₆₃
Hall	C₇×C₉ ⋊ D₁₈₅
Hall	C₅×C₉ ⋊ D₂₅₉
Sylow	C₃₇ ⋊ D₃₁₅
Hall	C₅×C₇ ⋊ D₃₃₃
Sylow	C₉ ⋊ D₁₂₉₅
Sylow	C₇ ⋊ D₁₆₆₅
Sylow	C₅ ⋊ D₂₃₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₇×C₉×C₃₇	C₂
Frattini	C₃	D₃₈₈₅
socle	C₃×C₅×C₇×C₃₇	D₃
LCS	G ▷ C₅×C₇×C₉×C₃₇
UCS	C₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 58
transitive degree	μ_t(G) = 11655

A minimal faithful permutation representation — 2 generators on 58 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);(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(16,21)(17,20)(18,19)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)

Polycyclic presentation:

2,3,5,7,37,3|b^3=f^2,b^a=b^2f,c^a=c^4,d^a=d^6,e^a=e^36,f^a=f^2

Conjugacy classes & character table

5829 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23310, 20);

D11655 #37552001