C₂²×#12874 #72310834

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#12874

Subgroups

	subgroup	G/H
center, socle	C₂³	#1823
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂³×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

7321 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#500001, C₂²×#494058, C₂²×#500351
maximal normal subgroup of	C₂²×#500001, C₂²×#494058, C₂²×#500351

GAP

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

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

C22×#12874 #72310834