D₉₂₇₅ #32915367

order	18550
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₂₇₅
unambiguous semidirect product	C₇⋊D₁₃₂₅
unambiguous semidirect product	C₂₅⋊D₃₇₁
unambiguous semidirect product	C₅₃⋊D₁₇₅

Splittings

Fitting, derived, Hall	C₇×C₂₅×C₅₃ ⋊ C₂
Hall	C₂₅×C₅₃ ⋊ D₇
Hall	C₇×C₅₃ ⋊ D₂₅
Hall	C₇×C₂₅ ⋊ D₅₃
Sylow	C₅₃ ⋊ D₁₇₅
Sylow	C₂₅ ⋊ D₃₇₁
Sylow	C₇ ⋊ D₁₃₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₂₅×C₅₃	C₂
Frattini	C₅	D₁₈₅₅
socle	C₅×C₇×C₅₃	D₅
LCS	G ▷ C₇×C₂₅×C₅₃
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 9275

A minimal faithful permutation representation — 2 generators on 85 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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,32)(28,31)(29,30)(34,85)(35,84)(36,83)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Polycyclic presentation:

2,5,7,53,5|b^5=e^4,b^a=b^4e,c^a=c^6,d^a=d^52,e^a=e^4

Conjugacy classes & character table

4639 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅₃×D₁₇₅, D₇×D₁₃₂₅, D₂₅×D₃₇₁, …
maximal normal subgroup of	D₅₃×D₁₇₅, D₇×D₁₃₂₅, D₂₅×D₃₇₁, …
maximal quotient of	Dic₉₂₇₅, C₂×D₉₂₇₅, D₂₇₈₂₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18550, 8);

D9275 #32915367