C₂²×#13004 #72309904

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

Aliases

direct product

C₂²×#13004

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂³×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ C₂⁵×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

12473 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#495721, C₂×#72250015, C₂×#72257070, …
maximal normal subgroup of	C₂²×#495721, C₂×#72250015, C₂×#72257070, …
maximal quotient of	C₂²×#136613

GAP

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

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

C22×#13004 #72309904