D₃×C₂₅×C₈₉ #27211161

order	13350
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₂₅×C₈₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₂₅×C₈₉
unambiguous semidirect product	C₃⋊(C₂×C₂₅×C₈₉)

Splittings

Fitting, Hall	C₃×C₂₅×C₈₉ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₂₅×C₈₉
Hall	C₃×C₈₉ ⋊ C₂×C₂₅
Hall	C₃×C₂₅ ⋊ C₂×C₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂₅×C₈₉	D₃
commutator, hypocenter	C₃	C₂×C₂₅×C₈₉
Frattini	C₅	C₅×D₃×C₈₉
Fitting	C₃×C₂₅×C₈₉	C₂
socle	C₃×C₅×C₈₉	C₂×C₅
prime radical	C₅×D₃×C₈₉	C₅
cosocle	C₃×C₅	C₂×C₅×C₈₉
LCS	G ▷ C₃
UCS	C₁ ◁ C₂₅×C₈₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 6675

A minimal faithful permutation representation — 2 generators on 117 points:

(2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92)(93,94,96,99,103,95,97,100,104,108,98,101,105,109,112,102,106,110,113,115,107,111,114,116,117);(1,2,3)

Conjugacy classes & character table

6675 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃×C₂₅×D₈₉, D₃×D₂₅×C₈₉, D₃×D₂₂₂₅, …
maximal normal subgroup of	D₃×C₂₅×D₈₉, D₃×D₂₅×C₈₉, D₃×D₂₂₂₅, …
maximal quotient of	Dic₃×C₂₅×C₈₉, C₂×D₃×C₂₅×C₈₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13350, 5);

D3×C25×C89 #27211161