C₃×C₄₃×Dic₂₅ #27166338

order	12900
isoclinism	isoclinic class of D₂₅
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₄₃ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×C₄₃×Dic₂₅

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄₃	D₂₅
commutator, hypocenter	C₂₅	C₃×C₄×C₄₃
Frattini	C₂×C₅	C₃×D₅×C₄₃
Fitting	C₂×C₃×C₂₅×C₄₃	C₂
socle, prime radical	C₂×C₃×C₅×C₄₃	D₅
cosocle	C₂×C₂₅	C₂×C₃×C₄₃
LCS	G ▷ C₂₅
UCS	C₁ ◁ C₂×C₃×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:

(2,4)(3,7)(5,10)(6,13)(8,16)(9,15)(11,12)(14,20)(17,21)(18,22)(19,24)(23,25)(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);(1,15,25,18,4,7,21,19,10,13,16,14,12,11,20,8,6,5,24,17,3,2,22,23,9)

Conjugacy classes & character table

3612 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 7);

C3×C43×Dic25 #27166338