C₂₅×#216 #11924853

order	1200
isoclinism	isoclinic class of #213
automorphism	?
homology	Abelianization (H₁) = C₂²×C₂₅ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₂₅×#216

Splittings

Hall	C₃×C₂₅ ⋊ Q₁₆
Sylow	C₃ ⋊ Q₁₆×C₂₅

Subgroups

	subgroup	G/H
center	C₂×C₂₅	#67
commutator	C₃×C₄	C₂²×C₂₅
Frattini	C₄×C₅	C₂×C₅×D₃
Fitting	C₃×Q₈×C₂₅	C₂
socle, prime radical	C₂×C₃×C₅	C₅×D₄
cosocle	C₃×C₄×C₅	C₂²×C₅
hypercenter	Q₈×C₂₅	D₃
hypocenter	C₃	Q₁₆×C₂₅
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₂₅ ◁ C₄×C₂₅ ◁ Q₈×C₂₅

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 1200

A minimal faithful permutation representation — 2 generators on 44 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)(29,31,33,39)(30,34,36,43)(32,41,42,38)(35,44,40,37);(27,28)(29,30,33,36)(31,37,39,44)(32,40,42,35)(34,41,43,38)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1200, 63);

C25×#216 #11924853