C₂₅×D₂₆₁ #27189374

order	13050
automorphism	?
homology	Abelianization (H₁) = C₂×C₂₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂₅×D₂₆₁
unambiguous semidirect product	C₉⋊(C₂₅×D₂₉)
unambiguous semidirect product	C₂₉⋊(D₉×C₂₅)

Splittings

Fitting, Hall	C₉×C₂₅×C₂₉ ⋊ C₂
derived, Hall	C₉×C₂₉ ⋊ C₂×C₂₅
Hall	C₂₅×C₂₉ ⋊ D₉
Hall	C₉×C₂₅ ⋊ D₂₉
Sylow	C₂₉ ⋊ D₉×C₂₅
Sylow	C₉ ⋊ C₂₅×D₂₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂₅	D₂₆₁
commutator, hypocenter	C₉×C₂₉	C₂×C₂₅
Frattini	C₃×C₅	C₅×D₈₇
Fitting	C₉×C₂₅×C₂₉	C₂
socle	C₃×C₅×C₂₉	C₅×D₃
prime radical	C₅×D₂₆₁	C₅
cosocle	C₅×C₉×C₂₉	C₂×C₅
LCS	G ▷ C₉×C₂₉
UCS	C₁ ◁ C₂₅

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 6525

A minimal faithful permutation representation — 2 generators on 63 points:

(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)(31,33)(32,36)(34,35)(37,38)(39,40,42,45,49,41,43,46,50,54,44,47,51,55,58,48,52,56,59,61,53,57,60,62,63);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(30,38,33,36,35,34,32,31,37)

Conjugacy classes & character table

3300 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉×C₂₅×D₂₉, D₂₅×D₂₆₁, D₂₂₅⋊D₂₉, …
maximal normal subgroup of	D₉×C₂₅×D₂₉, D₂₅×D₂₆₁, D₂₂₅⋊D₂₉, …
maximal quotient of	C₂₅×Dic₂₆₁, C₂×C₂₅×D₂₆₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13050, 6);

C25×D261 #27189374