Dic₂₅×C₄₃⋊C₃ #27166334

order	12900
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

direct product	Dic₂₅×C₄₃⋊C₃
unambiguous semidirect product	(C₄₃×Dic₂₅)⋊C₃

Splittings

derived, Hall	C₂₅×C₄₃ ⋊ C₃×C₄
Hall	C₄₃×Dic₂₅ ⋊ C₃
Hall	C₂₅×C₄₃⋊C₃ ⋊ C₄
Sylow	C₄₃ ⋊ C₃×Dic₂₅
Sylow	C₂₅ ⋊ C₄×C₄₃⋊C₃

Subgroups

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

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 4300

A minimal faithful permutation representation — 2 generators on 72 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)(54,55)(57,63)(60,64)(61,65)(62,67)(66,68)(69,70,71,72);(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)(44,58,68,61,47,50,64,62,53,56,59,57,55,54,63,51,49,48,67,60,46,45,65,66,52)

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, 3);

Dic25×C43⋊C3 #27166334