C₃×D₂₅×C₈₉ #27211159

order	13350
isoclinism	isoclinic class of D₂₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₂₅×C₈₉
unambiguous semidirect product	C₂₅⋊(C₂×C₃×C₈₉)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₈₉	D₂₅
commutator, cosocle, hypocenter	C₂₅	C₂×C₃×C₈₉
Frattini	C₅	C₃×D₅×C₈₉
Fitting	C₃×C₂₅×C₈₉	C₂
socle	C₃×C₅×C₈₉	D₅
LCS	G ▷ C₂₅
UCS	C₁ ◁ C₃×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:

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

Conjugacy classes & character table

3738 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₈₉×#1035, C₃×#22581809, C₃×D₂₅×D₈₉, …
maximal normal subgroup of	C₃×C₈₉×#1035, C₃×#22581809, C₃×D₂₅×D₈₉, …
maximal quotient of	C₃×C₈₉×Dic₂₅, C₂×C₃×D₂₅×C₈₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13350, 3);

C3×D25×C89 #27211159