C₅×D₄₁² #46777451

order	33620
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×D₄₁²

Splittings

Fitting, socle, Hall	C₅×C₄₁² ⋊ C₂²
derived, Sylow	C₄₁² ⋊ C₂²×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₄₁²
commutator, cosocle, hypocenter	C₄₁²	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₄₁²	C₂²
LCS	G ▷ C₄₁²
UCS	C₁ ◁ C₅

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 410

A minimal faithful permutation representation — 2 generators on 87 points:

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

Conjugacy classes & character table

2420 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33620, 86);

C5×D412 #46777451