C₇×C₁₉×#1035 #27210853

order	13300
isoclinism	isoclinic class of #1035
automorphism	?
homology	Abelianization (H₁) = C₄×C₇×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₇×C₁₉×#1035

Splittings

Fitting, Hall	C₇×C₁₉×C₂₅ ⋊ C₄
derived, Sylow	C₂₅ ⋊ C₄×C₇×C₁₉
Hall	C₁₉×C₂₅ ⋊ C₄×C₇
Hall	C₇×C₂₅ ⋊ C₄×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₇×C₁₉	#1035
commutator, hypocenter	C₂₅	C₄×C₇×C₁₉
Frattini	C₅	C₇×C₁₉×AGL(1,𝔽₅)
Fitting	C₇×C₁₉×C₂₅	C₄
socle	C₅×C₇×C₁₉	AGL(1,𝔽₅)
prime radical	C₇×C₁₉×D₂₅	C₂
cosocle	D₂₅	C₂×C₇×C₁₉
LCS	G ▷ C₂₅
UCS	C₁ ◁ C₇×C₁₉

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 3325

A minimal faithful permutation representation — 2 generators on 51 points:

(2,4,5,10)(3,8,9,20)(6,14,15,19)(7,17,18,11)(12,22,23,25)(13,16,21,24)(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);(1,2,6,16,13,3,7,4,11,22,8,19,12,23,14,20,25,17,10,18,9,21,24,15,5)

Conjugacy classes & character table

1330 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13300, 11);

C7×C19×#1035 #27210853