C₂²×#15950 #72337046

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂²×#15950

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

98885 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 24 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#687584, C₂²×#1418354, C₂²×#772779, …
maximal normal subgroup of	C₂²×#687584, C₂²×#1418354, C₂²×#772779, …

GAP

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

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

C22×#15950 #72337046