C₃×C₅²×D₈₉ #27211182

order	13350
isoclinism	isoclinic class of D₈₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅² · Schur multiplier (H₂) = C₅

Aliases

direct product	C₃×C₅²×D₈₉
unambiguous semidirect product	C₈₉⋊(C₂×C₃×C₅²)

Splittings

Fitting, socle, Hall	C₃×C₅²×C₈₉ ⋊ C₂
derived, Sylow	C₈₉ ⋊ C₂×C₃×C₅²
Hall	C₅²×C₈₉ ⋊ C₂×C₃
Hall	C₃×C₈₉ ⋊ C₂×C₅²

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅²	D₈₉
commutator, cosocle, hypocenter	C₈₉	C₂×C₃×C₅²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₈₉	C₂
LCS	G ▷ C₈₉
UCS	C₁ ◁ C₃×C₅²

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 102
transitive degree	μ_t(G) = 6675

A minimal faithful permutation representation — 2 generators on 102 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)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(95,96,97,98,99)(100,101,102);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(90,91,92,93,94)

Conjugacy classes & character table

3450 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₅²×#67063, C₃×C₅×#13184070, C₃×C₅×D₅×D₈₉, …
maximal normal subgroup of	C₃×C₅²×#67063, C₃×C₅×#13184070, C₃×C₅×D₅×D₈₉, …
maximal quotient of	C₃×C₅²×Dic₈₉, C₂×C₃×C₅²×D₈₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13350, 26);

C3×C52×D89 #27211182