C₂²×#47844 #72295946

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#47844

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

105637 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);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16);(5,6,7,8)(9,10,12,14)(11,15,16,13)(19,23)(21,24);(6,8)(11,16)(13,15)(17,18,20,22)(19,21,23,24);(9,10)(11,13)(12,14)(15,16)(17,19,20,23)(18,21,22,24);(5,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,24)(20,22)(21,23);(1,2)(3,4)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8062966, C₂²×#7850713, C₂²×#7837953
maximal normal subgroup of	C₂²×#8062966, C₂²×#7850713, C₂²×#7837953

GAP

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

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

C22×#47844 #72295946