C₂²×#72213827 #72291679

order	4096
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷

Aliases

	subgroup	G/H
center, socle	C₂⁴	#23900
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁶
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁶ ◁ C₂⁴×#120 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

At least 10187 conjugacy classes of subgroups; only the 5539 normal subgroups are shown.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 20 points:

(3,10)(6,12)(11,15)(13,16);(3,11)(10,15);(2,6,8,12)(9,13,14,16);(1,2,7,14)(3,12,15,13)(4,9,5,8)(6,11,16,10);(17,18);(19,20)

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

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,10)(6,12)(11,15)(13,16), (3,11)(10,15), (2,6,8,12)(9,13,14,16), (1,2,7,14)(3,12,15,13)(4,9,5,8)(6,11,16,10), (17,18), (19,20));