C₅²×D₃×C₈₃ #27159495

order	12450
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₅²×C₈₃ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₅²×D₃×C₈₃
unambiguous semidirect product	C₃⋊(C₂×C₅²×C₈₃)

Splittings

Fitting, socle, Hall	C₃×C₅²×C₈₃ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₅²×C₈₃
Hall	C₃×C₈₃ ⋊ C₂×C₅²
Hall	C₃×C₅² ⋊ C₂×C₈₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅²×C₈₃	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₅²×C₈₃
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₈₃	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₅²×C₈₃

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 6225

A minimal faithful permutation representation — 2 generators on 96 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)(92,93,94,95,96);(1,2,3)(87,88,89,90,91)

Conjugacy classes & character table

6225 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅²×D₃×D₈₃, C₅×D₃×D₅×C₈₃, C₅×D₃×D₄₁₅, …
maximal normal subgroup of	C₅²×D₃×D₈₃, C₅×D₃×D₅×C₈₃, C₅×D₃×D₄₁₅, …
maximal quotient of	C₅²×Dic₃×C₈₃, C₂×C₅²×D₃×C₈₃, C₅²×D₉×C₈₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12450, 21);

C52×D3×C83 #27159495