C₂²×#10941 #72342392

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂²×#10941

Subgroups

	subgroup	G/H
center	C₂²×C₄	#386
commutator	C₂³	C₂³×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂³	#1437
prime radical	C₂⁴×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

23809 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#368859
maximal normal subgroup of	C₂²×#368859

GAP

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

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

C22×#10941 #72342392