C₂×#30681140 #46777438

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

Aliases

direct product

C₂×#30681140

Splittings

Fitting, socle	C₂×C₄₁² ⋊ C₂×C₅
derived, Sylow	C₄₁² ⋊ C₂²×C₅
Hall	#22554261 ⋊ C₂²
Hall	C₂×C₄₁⋊D₄₁ ⋊ C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#30681140
commutator, cosocle, hypocenter	C₄₁²	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₄₁²	C₂×C₅
LCS	G ▷ C₄₁²
UCS	C₁ ◁ C₂

440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 3362

A minimal faithful permutation representation — 3 generators on 84 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33620, 73);

C2×#30681140 #46777438