D₁₆₂₂₅ #44570086

order	32450
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₂₂₅
unambiguous semidirect product	C₁₁⋊D₁₄₇₅
unambiguous semidirect product	C₂₅⋊D₆₄₉
unambiguous semidirect product	C₅₉⋊D₂₇₅

Splittings

Fitting, derived, Hall	C₁₁×C₂₅×C₅₉ ⋊ C₂
Hall	C₂₅×C₅₉ ⋊ D₁₁
Hall	C₁₁×C₅₉ ⋊ D₂₅
Hall	C₁₁×C₂₅ ⋊ D₅₉
Sylow	C₅₉ ⋊ D₂₇₅
Sylow	C₂₅ ⋊ D₆₄₉
Sylow	C₁₁ ⋊ D₁₄₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₁×C₂₅×C₅₉	C₂
Frattini	C₅	D₃₂₄₅
socle	C₅×C₁₁×C₅₉	D₅
LCS	G ▷ C₁₁×C₂₅×C₅₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 16225

A minimal faithful permutation representation — 2 generators on 95 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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,36)(28,35)(29,34)(30,33)(31,32)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)

Polycyclic presentation:

2,5,11,59,5|b^5=e^4,b^a=b^4e,c^a=c^10,d^a=d^58,e^a=e^4

Conjugacy classes & character table

8114 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₆₂₂₅
maximal normal subgroup of	C₂×D₁₆₂₂₅
maximal quotient of	Dic₁₆₂₂₅, C₂×D₁₆₂₂₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32450, 9);

D16225 #44570086