C₂×#30681143 #46777441

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

Aliases

direct product

C₂×#30681143

Splittings

Fitting, socle	C₂×C₄₁² ⋊ C₂×C₅
derived, Sylow	C₄₁² ⋊ C₂²×C₅
Hall	#22554262 ⋊ C₂²
Hall	C₂×C₄₁⋊D₄₁ ⋊ C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#30681143
commutator, cosocle, hypocenter	C₄₁²	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₄₁²	C₂×C₅
LCS	G ▷ C₄₁²
UCS	C₁ ◁ C₂

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 410

A minimal faithful permutation representation — 2 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)(45,49,50,65,68,46,47,55,57,67)(48,62,63,74,75,52,53,76,61,84)(51,71,72,73,66,59,60,82,79,56)(54,70,78,69,83,58,64,81,80,77);(1,2)(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)(44,45,48,54,55,62,72,83,76,84,57,69,81,64,60,73,68,66,47,51,61,74,59,49,56,67,79,71,70,78,80,65,75,63,77,82,53,50,58,52,46)

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, 76);

C2×#30681143 #46777441