C₂²×#12955 #72309871

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#12955

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂³×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

7881 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#495721, C₂×#72250110, C₂×#72257073, …
maximal normal subgroup of	C₂²×#495721, C₂×#72250110, C₂×#72257073, …
maximal quotient of	C₂²×#135908

GAP

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

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

C22×#12955 #72309871