D₁₃₄₅₅ #41002090

order	26910
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₄₅₅
unambiguous semidirect product	C₅⋊D₂₆₉₁
unambiguous semidirect product	C₉⋊D₁₄₉₅
unambiguous semidirect product	C₁₃⋊D₁₀₃₅
unambiguous semidirect product	C₂₃⋊D₅₈₅

Splittings

Fitting, derived, Hall	C₅×C₉×C₁₃×C₂₃ ⋊ C₂
Hall	C₉×C₁₃×C₂₃ ⋊ D₅
Hall	C₅×C₁₃×C₂₃ ⋊ D₉
Hall	C₅×C₉×C₂₃ ⋊ D₁₃
Hall	C₅×C₉×C₁₃ ⋊ D₂₃
Hall	C₁₃×C₂₃ ⋊ D₄₅
Hall	C₉×C₂₃ ⋊ D₆₅
Hall	C₉×C₁₃ ⋊ D₁₁₅
Hall	C₅×C₂₃ ⋊ D₁₁₇
Hall	C₅×C₁₃ ⋊ D₂₀₇
Hall	C₅×C₉ ⋊ D₂₉₉
Sylow	C₂₃ ⋊ D₅₈₅
Sylow	C₁₃ ⋊ D₁₀₃₅
Sylow	C₉ ⋊ D₁₄₉₅
Sylow	C₅ ⋊ D₂₆₉₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₉×C₁₃×C₂₃	C₂
Frattini	C₃	D₄₄₈₅
socle	C₃×C₅×C₁₃×C₂₃	D₃
LCS	G ▷ C₅×C₉×C₁₃×C₂₃
UCS	C₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 13455

A minimal faithful permutation representation — 2 generators on 50 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);(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)

Polycyclic presentation:

2,3,5,13,23,3|b^3=f^2,b^a=b^2f,c^a=c^4,d^a=d^12,e^a=e^22,f^a=f^2

Conjugacy classes & character table

6729 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₄₅₅
maximal normal subgroup of	C₂×D₁₃₄₅₅
maximal quotient of	Dic₁₃₄₅₅, C₂×D₁₃₄₅₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26910, 17);

D13455 #41002090