C₂²×#13584 #72306447

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰×C₄

Aliases

direct product

C₂²×#13584

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#353
commutator	C₄²	C₂⁴×C₄
Frattini, cosocle	#127	C₂⁵
prime radical	C₂²×#3151	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂³×#127 ◁ G
LECS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

10029 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 20 points:

(5,6,8,13,9,11,18,12)(7,14,15,19,16,20,17,10);(5,7)(6,10,12,14)(8,17,18,15)(9,16)(11,19,13,20);(1,2);(3,4);(1,2)(3,4)(6,11)(10,19)(12,13)(14,20)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#512862, C₂²×#152191, C₂²×#152049
maximal normal subgroup of	C₂²×#512862, C₂²×#152191, C₂²×#152049
maximal quotient of	#72338454

GAP

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

gap> G := Group((5,6,8,13,9,11,18,12)(7,14,15,19,16,20,17,10), (5,7)(6,10,12,14)(8,17,18,15)(9,16)(11,19,13,20), (1,2), (3,4), (1,2)(3,4)(6,11)(10,19)(12,13)(14,20));

C22×#13584 #72306447