C₂₅×D₂₇ #13062442

order	1350
isoclinism	isoclinic class of D₂₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₂₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂₅×D₂₇
unambiguous semidirect product	C₂₇⋊(C₂×C₂₅)

Splittings

Fitting, Hall	C₂₅×C₂₇ ⋊ C₂
derived, Sylow	C₂₇ ⋊ C₂×C₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂₅	D₂₇
commutator, hypocenter	C₂₇	C₂×C₂₅
Frattini	C₅×C₉	C₅×D₃
Fitting	C₂₅×C₂₇	C₂
socle	C₃×C₅	C₅×D₉
prime radical	C₅×D₂₇	C₅
cosocle	C₅×C₂₇	C₂×C₅
LCS	G ▷ C₂₇
UCS	C₁ ◁ C₂₅

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 675

A minimal faithful permutation representation — 2 generators on 52 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52);(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1350, 1);

C25×D27 #13062442