C₂²×#12981 #72311671

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂²×#12981

Subgroups

	subgroup	G/H
center	C₂²×C₈	#120
commutator	C₂³	C₂⁴×C₈
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂³	#1869
prime radical	C₂²×#525	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

10217 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2);(1,2)(3,4)(7,11)(15,21)(16,22)(17,23)(27,31)(28,32)(29,33)(35,36);(3,4);(1,2)(3,4)(5,6,10,14)(7,11,15,21)(8,12,19,25)(9,13,20,26)(16,22,27,31)(17,23,28,32)(18,24,30,34)(29,33,35,36);(5,7,8,16,9,17,18,29,10,15,19,27,20,28,30,35)(6,11,12,22,13,23,24,33,14,21,25,31,26,32,34,36)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#500351, C₂²×#500003, C₂²×#495505, …
maximal normal subgroup of	C₂²×#500351, C₂²×#500003, C₂²×#495505, …

GAP

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

gap> G := Group((1,2), (1,2)(3,4)(7,11)(15,21)(16,22)(17,23)(27,31)(28,32)(29,33)(35,36), (3,4), (1,2)(3,4)(5,6,10,14)(7,11,15,21)(8,12,19,25)(9,13,20,26)(16,22,27,31)(17,23,28,32)(18,24,30,34)(29,33,35,36), (5,7,8,16,9,17,18,29,10,15,19,27,20,28,30,35)(6,11,12,22,13,23,24,33,14,21,25,31,26,32,34,36));

C22×#12981 #72311671