C₂×D₂₅×C₄₃⋊C₃ #27166350

order	12900
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×D₂₅×C₄₃⋊C₃
unambiguous semidirect product	C₂₅⋊(C₂²×C₄₃⋊C₃)
unambiguous semidirect product	(C₂×C₂₅)⋊(C₂×C₄₃⋊C₃)
unambiguous semidirect product	(C₂×C₄₃×D₂₅)⋊C₃

Splittings

Fitting	C₂×C₂₅×C₄₃ ⋊ C₂×C₃
derived, Hall	C₂₅×C₄₃ ⋊ C₂²×C₃
Hall	C₂×C₄₃×D₂₅ ⋊ C₃
Hall	C₂₅×C₄₃⋊C₃ ⋊ C₂²
Sylow	C₄₃ ⋊ C₂×C₃×D₂₅
Sylow	C₂₅ ⋊ C₂²×C₄₃⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₂₅×C₄₃⋊C₃
commutator, cosocle, hypocenter	C₂₅×C₄₃	C₂²×C₃
Frattini	C₅	C₂×D₅×C₄₃⋊C₃
Fitting	C₂×C₂₅×C₄₃	C₂×C₃
socle	C₂×C₅×C₄₃	C₃×D₅
LCS	G ▷ C₂₅×C₄₃
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 70
transitive degree	μ_t(G) = 2150

A minimal faithful permutation representation — 2 generators on 70 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,33)(18,26,31)(21,29,28)(23,32,25)(27,36,40)(30,37,39)(34,41,35)(38,42,43)(45,47)(46,50)(48,53)(49,56)(51,59)(52,58)(54,61)(55,63)(57,65)(60,66)(62,67)(64,68);(1,2,4,8,15,20,5,9,16,25,35,32,12,19,28,24,34,41,22,31,39,36,14,23,33,40,43,42,26,7,13,21,30,38,37,17,3,6,11,18,27,29,10)(44,49,64,63,47,46,57,67,53,52,51,66,61,54,60,59,58,48,62,65,50,45,55,68,56)(69,70)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 19);

C2×D25×C43⋊C3 #27166350