D₉×D₃₆₅ #27190207

order	13140
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

direct product	D₉×D₃₆₅
unambiguous semidirect product	(C₅×D₉)⋊D₇₃
unambiguous semidirect product	(D₉×C₇₃)⋊D₅

Splittings

Fitting, derived, Hall	C₅×C₉×C₇₃ ⋊ C₂²
Hall	C₉×C₇₃ ⋊ C₂×D₅
Hall	C₅×C₇₃ ⋊ C₂×D₉
Sylow	C₇₃ ⋊ D₅×D₉
Hall	C₅×C₉ ⋊ C₂×D₇₃
Sylow	C₉ ⋊ C₂×D₃₆₅
Sylow	C₅ ⋊ D₉×D₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₉×C₇₃	C₂²
Frattini	C₃	D₃×D₃₆₅
socle	C₃×C₅×C₇₃	C₂×D₃
LCS	G ▷ C₅×C₉×C₇₃
UCS	C₁

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 3285

A minimal faithful permutation representation — 2 generators on 87 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)(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)(75,76)(77,78)(79,84,82,81,86,83,85,80,87);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(74,75,77,78,76)(80,82)(81,85)(83,86)(84,87)

Conjugacy classes & character table

1104 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 41);

D9×D365 #27190207