C₂²×#10757 #72342398

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂²×#10757

Subgroups

	subgroup	G/H
center	C₂²×C₄	#386
commutator	C₂³	C₂³×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂³	C₂²⋊#95
prime radical	C₂⁴×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

38353 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#394600
maximal normal subgroup of	C₂²×#394600

GAP

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

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

C22×#10757 #72342398