D₈₇₄₅ #31775822

order	17490
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₈₇₄₅
unambiguous semidirect product	C₃⋊D₂₉₁₅
unambiguous semidirect product	C₅⋊D₁₇₄₉
unambiguous semidirect product	C₁₁⋊D₇₉₅
unambiguous semidirect product	C₅₃⋊D₁₆₅

Splittings

Fitting, socle, derived, Hall	C₃×C₅×C₁₁×C₅₃ ⋊ C₂
Hall	C₅×C₁₁×C₅₃ ⋊ D₃
Hall	C₃×C₁₁×C₅₃ ⋊ D₅
Hall	C₃×C₅×C₅₃ ⋊ D₁₁
Hall	C₁₁×C₅₃ ⋊ D₁₅
Hall	C₅×C₅₃ ⋊ D₃₃
Hall	C₃×C₅×C₁₁ ⋊ D₅₃
Hall	C₃×C₅₃ ⋊ D₅₅
Hall	C₅×C₁₁ ⋊ D₁₅₉
Sylow	C₅₃ ⋊ D₁₆₅
Hall	C₃×C₁₁ ⋊ D₂₆₅
Hall	C₃×C₅ ⋊ D₅₈₃
Sylow	C₁₁ ⋊ D₇₉₅
Sylow	C₅ ⋊ D₁₇₄₉
Sylow	C₃ ⋊ D₂₉₁₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₁₁×C₅₃	C₂
LCS	G ▷ C₃×C₅×C₁₁×C₅₃
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 8745

A minimal faithful permutation representation — 2 generators on 72 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);(2,3)(5,8)(6,7)(10,19)(11,18)(12,17)(13,16)(14,15)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(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:

2,3,5,11,53|b^a=b^2,c^a=c^4,d^a=d^10,e^a=e^52

Conjugacy classes & character table

4374 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅₃×D₁₆₅, D₁₁×D₇₉₅, D₁₅×D₅₈₃, …
maximal normal subgroup of	D₅₃×D₁₆₅, D₁₁×D₇₉₅, D₁₅×D₅₈₃, …
maximal quotient of	Dic₈₇₄₅, C₂×D₈₇₄₅, D₂₆₂₃₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17490, 17);

D8745 #31775822