C₂₅×Dic₁₂₉ #27166341

order	12900
automorphism	?
homology	Abelianization (H₁) = C₄×C₂₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂₅×Dic₁₂₉
unambiguous semidirect product	C₃⋊(C₂₅×Dic₄₃)
unambiguous semidirect product	C₄₃⋊(Dic₃×C₂₅)

Splittings

derived, Hall	C₃×C₄₃ ⋊ C₄×C₂₅
Hall	C₃×C₂₅×C₄₃ ⋊ C₄
Hall	C₂₅×C₄₃ ⋊ Dic₃
Hall	C₃×C₂₅ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ Dic₃×C₂₅
Sylow	C₃ ⋊ C₂₅×Dic₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₂₅	D₁₂₉
commutator, hypocenter	C₃×C₄₃	C₄×C₂₅
Frattini	C₂×C₅	C₅×D₁₂₉
Fitting	C₂×C₃×C₂₅×C₄₃	C₂
socle, prime radical, cosocle	C₂×C₃×C₅×C₄₃	C₂×C₅
LCS	G ▷ C₃×C₄₃
UCS	C₁ ◁ C₂×C₂₅

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)(4,5)(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)(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);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(44,45,46)

Conjugacy classes & character table

3300 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 10);

C25×Dic129 #27166341