C₂²×#51068 #72236644

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂²×#51068

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

115365 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7792761, C₂²×#7850315, C₂³×#51068, …
maximal normal subgroup of	C₂²×#7792761, C₂²×#7850315, C₂³×#51068, …
maximal quotient of	C₂²×#1262353, C₂²×#1268761, C₂³×#51068, …

GAP

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

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

C22×#51068 #72236644