Dic₃×C₂₅×C₄₃ #27166340

order	12900
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₂₅×C₄₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	Dic₃×C₂₅×C₄₃
unambiguous semidirect product	C₃⋊(C₄×C₂₅×C₄₃)

Splittings

derived, Sylow	C₃ ⋊ C₄×C₂₅×C₄₃
Hall	C₃×C₂₅×C₄₃ ⋊ C₄
Hall	C₃×C₄₃ ⋊ C₄×C₂₅
Hall	C₃×C₂₅ ⋊ C₄×C₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₂₅×C₄₃	D₃
commutator, hypocenter	C₃	C₄×C₂₅×C₄₃
Frattini	C₂×C₅	C₅×D₃×C₄₃
Fitting	C₂×C₃×C₂₅×C₄₃	C₂
socle, prime radical	C₂×C₃×C₅×C₄₃	C₂×C₅
cosocle	C₂×C₃×C₅	C₂×C₅×C₄₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₂₅×C₄₃

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 12900

A minimal faithful permutation representation — 2 generators on 75 points:

(2,3)(4,7,6,5)(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)(51,52,54,57,61,53,55,58,62,66,56,59,63,67,70,60,64,68,71,73,65,69,72,74,75);(1,2,3)

Conjugacy classes & character table

6450 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 9);

Dic3×C25×C43 #27166340