C₂²×#13040 #72309850

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂²×#13040

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂³×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

10609 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72256927, C₂²×#495721, C₂×#72250205, …
maximal normal subgroup of	C₂×#72256927, C₂²×#495721, C₂×#72250205, …
maximal quotient of	C₂²×#136707, C₂×#72286049

GAP

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

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

C22×#13040 #72309850