C₂²×#120×#127 #72320594

order	4096
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁸×C₄

Aliases

direct product

C₂²×#120×#127

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁸
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

24736683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 20 points:

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₂²×#120×#392

GAP

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

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

C22×#120×#127 #72320594