C₃×C₂₅×D₈₃ #27159476

order	12450
isoclinism	isoclinic class of D₈₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₂₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₂₅×D₈₃
unambiguous semidirect product	C₈₃⋊(C₂×C₃×C₂₅)

Splittings

Fitting, Hall	C₃×C₂₅×C₈₃ ⋊ C₂
derived, Sylow	C₈₃ ⋊ C₂×C₃×C₂₅
Hall	C₂₅×C₈₃ ⋊ C₂×C₃
Hall	C₃×C₈₃ ⋊ C₂×C₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₂₅	D₈₃
commutator, hypocenter	C₈₃	C₂×C₃×C₂₅
Frattini	C₅	C₃×C₅×D₈₃
Fitting	C₃×C₂₅×C₈₃	C₂
socle	C₃×C₅×C₈₃	C₂×C₅
prime radical	C₃×C₅×D₈₃	C₅
cosocle	C₅×C₈₃	C₂×C₃×C₅
LCS	G ▷ C₈₃
UCS	C₁ ◁ C₃×C₂₅

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 6225

A minimal faithful permutation representation — 2 generators on 111 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)(84,88,96,104,108,86,92,101,107,94,89,97,105,90,99,93,102,87,95,103,98,85,91,100,106)(109,110,111);(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)

Conjugacy classes & character table

3225 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×D₂₅×D₈₃, D₃×C₂₅×D₈₃, D₇₅×D₈₃, …
maximal normal subgroup of	C₃×D₂₅×D₈₃, D₃×C₂₅×D₈₃, D₇₅×D₈₃, …
maximal quotient of	C₃×C₂₅×Dic₈₃, C₂×C₃×C₂₅×D₈₃, C₉×C₂₅×D₈₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12450, 2);

C3×C25×D83 #27159476