C₂²×#12948 #72320099

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

Aliases

direct product

C₂²×#12948

Subgroups

	subgroup	G/H
center, socle	C₂³	#1864
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂³×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

7473 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:

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

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×#72319959

GAP

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

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

C22×#12948 #72320099