D₁₈₆₅₅ #49152437

order	37310
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₈₆₅₅
unambiguous semidirect product	C₅⋊D₃₇₃₁
unambiguous semidirect product	C₇⋊D₂₆₆₅
unambiguous semidirect product	C₁₃⋊D₁₄₃₅
unambiguous semidirect product	C₄₁⋊D₄₅₅

Splittings

Fitting, socle, derived, Hall	C₅×C₇×C₁₃×C₄₁ ⋊ C₂
Hall	C₇×C₁₃×C₄₁ ⋊ D₅
Hall	C₅×C₁₃×C₄₁ ⋊ D₇
Hall	C₅×C₇×C₄₁ ⋊ D₁₃
Hall	C₁₃×C₄₁ ⋊ D₃₅
Hall	C₅×C₇×C₁₃ ⋊ D₄₁
Hall	C₇×C₄₁ ⋊ D₆₅
Hall	C₅×C₄₁ ⋊ D₉₁
Hall	C₇×C₁₃ ⋊ D₂₀₅
Hall	C₅×C₁₃ ⋊ D₂₈₇
Sylow	C₄₁ ⋊ D₄₅₅
Hall	C₅×C₇ ⋊ D₅₃₃
Sylow	C₁₃ ⋊ D₁₄₃₅
Sylow	C₇ ⋊ D₂₆₆₅
Sylow	C₅ ⋊ D₃₇₃₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₇×C₁₃×C₄₁	C₂
LCS	G ▷ C₅×C₇×C₁₃×C₄₁
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 18655

A minimal faithful permutation representation — 2 generators on 66 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);(2,5)(3,4)(7,12)(8,11)(9,10)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Polycyclic presentation:

2,5,7,13,41|b^a=b^4,c^a=c^6,d^a=d^12,e^a=e^40

Conjugacy classes & character table

9329 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37310, 17);

D18655 #49152437