C₂²×#45977 #72324835

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷×C₄²

Aliases

direct product

C₂²×#45977

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

111621 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 24 points:

(6,8)(9,10,14,15)(11,16,12,13)(19,23)(22,24);(10,15)(11,12)(18,21)(19,23);(9,11,14,12)(10,13,15,16)(17,18)(19,24)(20,21)(22,23);(3,4);(6,8)(9,12,14,11)(10,13,15,16)(17,19,20,23)(18,22,21,24);(1,2)(3,4);(5,6)(7,8)(9,13)(10,12)(11,15)(14,16)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7783989
maximal normal subgroup of	C₂²×#7783989

GAP

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

gap> G := Group((6,8)(9,10,14,15)(11,16,12,13)(19,23)(22,24), (10,15)(11,12)(18,21)(19,23), (9,11,14,12)(10,13,15,16)(17,18)(19,24)(20,21)(22,23), (3,4), (6,8)(9,12,14,11)(10,13,15,16)(17,19,20,23)(18,22,21,24), (1,2)(3,4), (5,6)(7,8)(9,13)(10,12)(11,15)(14,16));

C22×#45977 #72324835