C₉×D₇₂₅ #27189372

order	13050
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₉×D₇₂₅
unambiguous semidirect product	C₂₅⋊(C₉×D₂₉)
unambiguous semidirect product	C₂₉⋊(C₉×D₂₅)

Splittings

Fitting, Hall	C₉×C₂₅×C₂₉ ⋊ C₂
derived, Hall	C₂₅×C₂₉ ⋊ C₂×C₉
Hall	C₉×C₂₉ ⋊ D₂₅
Hall	C₉×C₂₅ ⋊ D₂₉
Sylow	C₂₉ ⋊ C₉×D₂₅
Sylow	C₂₅ ⋊ C₉×D₂₉

Subgroups

	subgroup	G/H
center, hypercenter	C₉	D₇₂₅
commutator, hypocenter	C₂₅×C₂₉	C₂×C₉
Frattini	C₃×C₅	C₃×D₁₄₅
Fitting	C₉×C₂₅×C₂₉	C₂
socle	C₃×C₅×C₂₉	C₃×D₅
prime radical	C₃×D₇₂₅	C₃
cosocle	C₃×C₂₅×C₂₉	C₂×C₃
LCS	G ▷ C₂₅×C₂₉
UCS	C₁ ◁ C₉

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)(34,39)(35,42)(37,45)(38,44)(40,41)(43,49)(46,50)(47,51)(48,53)(52,54)(55,56,58,57,59,61,60,62,63);(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)(30,44,54,47,33,36,50,48,39,42,45,43,41,40,49,37,35,34,53,46,32,31,51,52,38)

Conjugacy classes & character table

3276 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₉×#15668503, C₉×#15668502, C₉⋊#15668503, …
maximal normal subgroup of	C₉×#15668503, C₉×#15668502, C₉⋊#15668503, …
maximal quotient of	C₉×Dic₇₂₅, C₂×C₉×D₇₂₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13050, 4);

C9×D725 #27189372