C₂²×#13291 #72311447

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂²×#13291

Subgroups

	subgroup	G/H
center, socle	C₂³	#1993
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂⁵	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5277 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#500940, C₂²×#495573, C₂²×#500473
maximal normal subgroup of	C₂²×#500940, C₂²×#495573, C₂²×#500473

GAP

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

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

C22×#13291 #72311447