C₂²×#7631334 #72324184

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶×C₄

Aliases

direct product

C₂²×#7631334

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

2471296 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×D₄×#3484
maximal normal subgroup of	C₂²×D₄×#3484

GAP

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

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

C22×#7631334 #72324184