C₂²×#13459 #72308190

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#13459

Subgroups

	subgroup	G/H
center	C₂²×C₄	#498
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂³	#2058
prime radical	C₂⁵	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5493 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);(5,6)(7,16,19,30)(8,14)(9,23)(10,25)(11,12)(13,20,26,34)(15,24)(17,31,21,36)(18,33)(22,27)(28,35,29,32);(3,4)(6,12)(13,26)(14,25)(15,27)(16,30)(23,33)(28,29)(31,36);(6,12)(7,17,19,21)(9,18)(13,28,26,29)(14,25)(16,31,30,36)(20,35,34,32)(22,24);(5,7,18,34,10,21,22,32,11,19,9,20,8,17,24,35)(6,13,27,30,14,29,33,36,12,26,15,16,25,28,23,31)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72307107, C₂×#72307111, C₂²×#500349, …
maximal normal subgroup of	C₂×#72307107, C₂×#72307111, C₂²×#500349, …

GAP

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

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

C22×#13459 #72308190