C₅²⋊C₃×D₈₃ #27159488

order	12450
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₅²⋊C₃×D₈₃
unambiguous semidirect product	C₈₃⋊(C₂×C₅²⋊C₃)
unambiguous semidirect product	(C₅²×D₈₃)⋊C₃

Splittings

Fitting, socle, derived, Hall	C₅²×C₈₃ ⋊ C₂×C₃
Hall	C₅²⋊C₃×C₈₃ ⋊ C₂
Hall	C₅²×D₈₃ ⋊ C₃
Sylow	C₈₃ ⋊ C₂×C₅²⋊C₃
Sylow	C₅² ⋊ C₃×D₈₃

Subgroups

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

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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#3874×D₈₃, (C₅⋊D₅)⋊C₃×D₈₃, C₂×C₅²⋊C₃×D₈₃, …
maximal normal subgroup of	#3874×D₈₃, (C₅⋊D₅)⋊C₃×D₈₃, C₂×C₅²⋊C₃×D₈₃, …
maximal quotient of	C₅²⋊C₃×Dic₈₃, C₂×C₅²⋊C₃×D₈₃, D₈₃×C₅²⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12450, 14);

C52⋊C3×D83 #27159488