C₂₅×#235 #11924952

order	1200
isoclinism	isoclinic class of Dic₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₅×#235

Splittings

Fitting	C₂×C₃×C₄×C₂₅ ⋊ C₂
Hall	C₃×C₂₅ ⋊ #41
Sylow	C₃ ⋊ #41×C₂₅

Subgroups

	subgroup	G/H
center	C₄×C₂₅	C₂×D₃
commutator	C₂×C₃	C₂³×C₂₅
Frattini	C₂×C₅	C₂²×C₅×D₃
Fitting	C₂×C₃×C₄×C₂₅	C₂
socle, cosocle	C₂×C₃×C₅	C₂³×C₅
prime radical	C₅×#235	C₅
hypercenter	C₂×C₄×C₂₅	D₃
hypocenter	C₃	#41×C₂₅
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₂₅ ◁ C₂×C₄×C₂₅

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 600

A minimal faithful permutation representation — 3 generators on 36 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)(30,34)(33,36);(29,31,32,35)(30,33,34,36);(27,28)(29,30)(31,33)(32,34)(35,36)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1200, 162);

C25×#235 #11924952