C₂³×#7630939 #72291080

order	4096
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁹ · Schur multiplier (H₂) = C₂³⁶

Aliases

direct product

C₂³×#7630939

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁹
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

250028539 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 9 generators on 22 points:

(15,16,19,18)(17,21,22,20);(3,4)(15,17)(16,20)(18,21)(19,22);(8,10)(12,14);(12,14)(15,16,19,18)(17,21,22,20);(16,18)(20,21);(15,17)(16,21)(18,20)(19,22);(7,8)(9,10)(11,12)(13,14)(15,18,19,16)(17,21,22,20);(5,6);(1,2)(3,4)(5,6)

Conjugacy classes & character table

1024 conjugacy classes. The character table is too large to display.

Relationships

automorphism group of

#17971, #17990

GAP

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

gap> G := Group((15,16,19,18)(17,21,22,20), (3,4)(15,17)(16,20)(18,21)(19,22), (8,10)(12,14), (12,14)(15,16,19,18)(17,21,22,20), (16,18)(20,21), (15,17)(16,21)(18,20)(19,22), (7,8)(9,10)(11,12)(13,14)(15,18,19,16)(17,21,22,20), (5,6), (1,2)(3,4)(5,6));

C23×#7630939 #72291080