C₂²×#12752 #72320137

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂²×#12752

Subgroups

	subgroup	G/H
center, socle	C₂³	#1786
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂²×#3584	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

30945 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#495737
maximal normal subgroup of	C₂²×#495737
maximal quotient of	C₂×#72319973

GAP

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

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

C22×#12752 #72320137