C₂²×#39334 #72324631

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂²×#39334

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

129401 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 20 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7916101
maximal normal subgroup of	C₂²×#7916101

GAP

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

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

C22×#39334 #72324631