C₁₉×#11761507 #34117807

order	20140
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₁₉×#11761507

Splittings

Fitting, socle, Hall	C₅×C₁₉×C₅₃ ⋊ C₄
derived, Hall	C₅×C₅₃ ⋊ C₄×C₁₉
Hall	C₁₉×C₅₃ ⋊ AGL(1,𝔽₅)
Hall	C₅×C₁₉ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ C₁₉×AGL(1,𝔽₅)
Sylow	C₅ ⋊ C₁₉×Dic₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₉	#11761507
commutator, hypocenter	C₅×C₅₃	C₄×C₁₉
Frattini	C₁	G
Fitting, socle	C₅×C₁₉×C₅₃	C₄
prime radical	D₅×C₁₉×C₅₃	C₂
cosocle	D₅×C₅₃	C₂×C₁₉
LCS	G ▷ C₅×C₅₃
UCS	C₁ ◁ C₁₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 5035

A minimal faithful permutation representation — 2 generators on 77 points:

(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)(52,53)(55,58,57,56)(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,55,56,58,57)

Conjugacy classes & character table

2071 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20140, 16);

C19×#11761507 #34117807