D₉₇₇₅ #34086371

order	19550
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₇₇₅
unambiguous semidirect product	C₁₇⋊D₅₇₅
unambiguous semidirect product	C₂₃⋊D₄₂₅
unambiguous semidirect product	C₂₅⋊D₃₉₁

Splittings

Fitting, derived, Hall	C₁₇×C₂₃×C₂₅ ⋊ C₂
Hall	C₂₃×C₂₅ ⋊ D₁₇
Hall	C₁₇×C₂₅ ⋊ D₂₃
Hall	C₁₇×C₂₃ ⋊ D₂₅
Sylow	C₂₅ ⋊ D₃₉₁
Sylow	C₂₃ ⋊ D₄₂₅
Sylow	C₁₇ ⋊ D₅₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₇×C₂₃×C₂₅	C₂
Frattini	C₅	D₁₉₅₅
socle	C₅×C₁₇×C₂₃	D₅
LCS	G ▷ C₁₇×C₂₃×C₂₅
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 9775

A minimal faithful permutation representation — 2 generators on 65 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);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)

Polycyclic presentation:

2,5,17,23,5|b^5=e^4,b^a=b^4e,c^a=c^16,d^a=d^22,e^a=e^4

Conjugacy classes & character table

4889 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₉₃₂₅
maximal quotient of	D₂₉₃₂₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19550, 8);

D9775 #34086371