C₅×C₁₃×D₉₉ #27166019

order	12870
isoclinism	isoclinic class of D₉₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₁₃×D₉₉
unambiguous semidirect product	C₉⋊(C₅×C₁₃×D₁₁)

Splittings

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₁₃	D₉₉
commutator, cosocle, hypocenter	C₉×C₁₁	C₂×C₅×C₁₃
Frattini	C₃	C₅×C₁₃×D₃₃
Fitting	C₅×C₉×C₁₁×C₁₃	C₂
socle	C₃×C₅×C₁₁×C₁₃	D₃
LCS	G ▷ C₉×C₁₁
UCS	C₁ ◁ C₅×C₁₃

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 6435

A minimal faithful permutation representation — 2 generators on 38 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(13,15)(14,18)(16,17)(19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33)(34,35,36,37,38);(1,2,4,6,8,10,11,9,7,5,3)(12,20,15,18,17,16,14,13,19)

Conjugacy classes & character table

3315 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×D₁₃×D₉₉, C₅×C₁₃×D₉×D₁₁, C₅×D₉₉⋊D₁₃, …
maximal normal subgroup of	C₅×D₁₃×D₉₉, C₅×C₁₃×D₉×D₁₁, C₅×D₉₉⋊D₁₃, …
maximal quotient of	C₅×C₁₃×Dic₉₉, C₂×C₅×C₁₃×D₉₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12870, 14);

Fitting, Hall	C₅×C₉×C₁₁×C₁₃ ⋊ C₂
derived, Hall	C₉×C₁₁ ⋊ C₂×C₅×C₁₃
Hall	C₉×C₁₁×C₁₃ ⋊ C₂×C₅
Hall	C₅×C₁₁×C₁₃ ⋊ D₉
Hall	C₅×C₉×C₁₃ ⋊ D₁₁
Hall	C₅×C₉×C₁₁ ⋊ C₂×C₁₃
Hall	C₁₁×C₁₃ ⋊ C₅×D₉
Hall	C₉×C₁₃ ⋊ C₅×D₁₁
Hall	C₅×C₁₁ ⋊ C₁₃×D₉
Hall	C₅×C₉ ⋊ C₁₃×D₁₁
Sylow	C₁₁ ⋊ C₅×C₁₃×D₉
Sylow	C₉ ⋊ C₅×C₁₃×D₁₁

C5×C13×D99 #27166019