C₈₃×(C₅⋊D₅)⋊C₃ #27159486

order	12450
isoclinism	isoclinic class of (C₅⋊D₅)⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈₃ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₈₃×(C₅⋊D₅)⋊C₃
unambiguous semidirect product	(C₅⋊D₅)⋊(C₃×C₈₃)
unambiguous semidirect product	(C₅⋊D₅×C₈₃)⋊C₃

Splittings

Fitting, socle, Hall	C₅²×C₈₃ ⋊ C₂×C₃
derived, Sylow	C₅² ⋊ C₂×C₃×C₈₃
Hall	C₅²⋊C₃×C₈₃ ⋊ C₂
Hall	C₅⋊D₅×C₈₃ ⋊ C₃
Hall	C₅²⋊C₃ ⋊ C₂×C₈₃
Hall	C₅⋊D₅ ⋊ C₃×C₈₃

Subgroups

	subgroup	G/H
center, hypercenter	C₈₃	(C₅⋊D₅)⋊C₃
commutator, cosocle, hypocenter	C₅²	C₂×C₃×C₈₃
Frattini	C₁	G
Fitting, socle	C₅²×C₈₃	C₂×C₃
LCS	G ▷ C₅²
UCS	C₁ ◁ C₈₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 98
transitive degree	μ_t(G) = 1245

A minimal faithful permutation representation — 2 generators on 98 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₈₃×#3874⋊C₂, ((C₅⋊D₅)⋊C₃)⋊D₈₃, C₈₃×#64480, …
maximal normal subgroup of	C₈₃×#3874⋊C₂, ((C₅⋊D₅)⋊C₃)⋊D₈₃, C₈₃×#64480, …
maximal quotient of	C₈₃×#1039⋊C₃, C₂×C₈₃×(C₅⋊D₅)⋊C₃, C₈₃×(C₅⋊D₅)⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12450, 12);

C83×(C5⋊D5)⋊C3 #27159486