D₁₅×C₈₉ #14581908

order	2670
isoclinism	isoclinic class of D₁₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₈₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₁₅×C₈₉
unambiguous semidirect product	C₃⋊(D₅×C₈₉)
unambiguous semidirect product	C₅⋊(D₃×C₈₉)

Splittings

Fitting, socle, Hall	C₃×C₅×C₈₉ ⋊ C₂
derived, Hall	C₃×C₅ ⋊ C₂×C₈₉
Hall	C₅×C₈₉ ⋊ D₃
Hall	C₃×C₈₉ ⋊ D₅
Sylow	C₅ ⋊ D₃×C₈₉
Sylow	C₃ ⋊ D₅×C₈₉

Subgroups

	subgroup	G/H
center, hypercenter	C₈₉	D₁₅
commutator, cosocle, hypocenter	C₃×C₅	C₂×C₈₉
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₈₉	C₂
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₈₉

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 1335

A minimal faithful permutation representation — 2 generators on 97 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₈₉×D₇₅, C₈₉×C₃⋊(C₅⋊D₅), C₅×D₁₅×C₈₉, …
maximal normal subgroup of	C₅×D₁₅×C₈₉, C₇×D₁₅×C₈₉, C₁₁×D₁₅×C₈₉, …
maximal quotient of	C₈₉×D₇₅, C₈₉×C₃⋊(C₅⋊D₅), C₅×D₁₅×C₈₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2670, 7);

D15×C89 #14581908