C₅²×Dic₁₂₉ #27166395

order	12900
automorphism	?
homology	Abelianization (H₁) = C₄×C₅² · Schur multiplier (H₂) = C₅

Aliases

direct product	C₅²×Dic₁₂₉
unambiguous semidirect product	C₃⋊(C₅²×Dic₄₃)
unambiguous semidirect product	C₄₃⋊(C₅²×Dic₃)

Splittings

derived, Hall	C₃×C₄₃ ⋊ C₄×C₅²
Hall	C₃×C₅²×C₄₃ ⋊ C₄
Hall	C₅²×C₄₃ ⋊ Dic₃
Hall	C₃×C₅² ⋊ Dic₄₃
Sylow	C₄₃ ⋊ C₅²×Dic₃
Sylow	C₃ ⋊ C₅²×Dic₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅²	D₁₂₉
commutator, hypocenter	C₃×C₄₃	C₄×C₅²
Frattini	C₂	C₅²×D₁₂₉
Fitting, socle, prime radical	C₂×C₃×C₅²×C₄₃	C₂
cosocle	C₂×C₃×C₄₃	C₂×C₅²
LCS	G ▷ C₃×C₄₃
UCS	C₁ ◁ C₂×C₅²

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 12900

A minimal faithful permutation representation — 2 generators on 60 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)(56,57,58,59,60);(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)(51,52,53,54,55)

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, 64);

C52×Dic129 #27166395