C₅×C₁₉⋊#6157 #34117802

order	20140
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₁₉⋊#6157
unambiguous semidirect product	C₁₉⋊(C₅×#6157)

Splittings

Fitting, socle, Hall	C₅×C₁₉×C₅₃ ⋊ C₄
derived, Hall	C₁₉×C₅₃ ⋊ C₄×C₅
Hall	C₅×C₅₃ ⋊ Dic₁₉
Hall	C₅×C₁₉ ⋊ #6157
Sylow	C₅₃ ⋊ C₅×Dic₁₉
Sylow	C₁₉ ⋊ C₅×#6157

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₁₉⋊#6157
commutator, hypocenter	C₁₉×C₅₃	C₄×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₁₉×C₅₃	C₄
prime radical	C₅×C₁₉×D₅₃	C₂
cosocle	C₁₉×D₅₃	C₂×C₅
LCS	G ▷ C₁₉×C₅₃
UCS	C₁ ◁ C₅

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:

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

Conjugacy classes & character table

1345 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20140, 11);

C5×C19⋊#6157 #34117802