D₅₅₂₅ #26010753

order	11050
automorphism	Aut(G) order 21216000
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₅₅₂₅
unambiguous semidirect product	C₁₃⋊D₄₂₅
unambiguous semidirect product	C₁₇⋊D₃₂₅
unambiguous semidirect product	C₂₅⋊D₂₂₁

Splittings

Fitting, derived, Hall	C₁₃×C₁₇×C₂₅ ⋊ C₂
Hall	C₁₇×C₂₅ ⋊ D₁₃
Hall	C₁₃×C₂₅ ⋊ D₁₇
Hall	C₁₃×C₁₇ ⋊ D₂₅
Sylow	C₂₅ ⋊ D₂₂₁
Sylow	C₁₇ ⋊ D₃₂₅
Sylow	C₁₃ ⋊ D₄₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₃×C₁₇×C₂₅	C₂
Frattini	C₅	D₁₁₀₅
socle	C₅×C₁₃×C₁₇	D₅
LCS	G ▷ C₁₃×C₁₇×C₂₅
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 5525

A minimal faithful permutation representation — 2 generators on 55 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);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Polycyclic presentation:

2,5,13,17,5|b^5=e^4,b^a=b^4e,c^a=c^12,d^a=d^16,e^a=e^4

Conjugacy classes & character table

2764 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#36348431, #36348432, #36348435, …
maximal normal subgroup of	#36348431, #36348432, #36348435, …
maximal quotient of	Dic₅₅₂₅, C₂×D₅₅₂₅, C₃×D₅₅₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(11050, 8);

D5525 #26010753