C₅×#20269608 #46777452

order	33620
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₅×#20269608

Splittings

Fitting, socle, Hall	C₅×C₄₁² ⋊ C₄
derived, Sylow	C₄₁² ⋊ C₄×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₅	#20269608
commutator, hypocenter	C₄₁²	C₄×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₄₁²	C₄
prime radical	C₅×C₄₁⋊D₄₁	C₂
cosocle	C₄₁⋊D₄₁	C₂×C₅
LCS	G ▷ C₄₁²
UCS	C₁ ◁ C₅

264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 8405

A minimal faithful permutation representation — 3 generators on 87 points:

(1,2,3,4,5)(7,8,9,11)(10,13,14,20)(12,17,18,16)(15,22,23,33)(19,26,27,29)(21,30,31,25)(24,35,36,40)(28,38,39,45)(32,42,43,41)(34,46,44,37)(48,49,50,52)(51,54,55,61)(53,58,59,57)(56,63,64,74)(60,67,68,70)(62,71,72,66)(65,76,77,81)(69,79,80,86)(73,83,84,82)(75,87,85,78);(47,48,51,56,65,76,80,68,59,52,57,66,78,79,63,75,84,72,61,70,82,83,67,54,62,73,85,74,86,87,71,58,49,53,60,69,81,77,64,55,50);(6,7,10,15,24,35,39,27,18,11,16,25,37,38,22,34,43,31,20,29,41,42,26,13,21,32,44,33,45,46,30,17,8,12,19,28,40,36,23,14,9)

Conjugacy classes & character table

2120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33620, 87);

C5×#20269608 #46777452