C₂²×#12202 #72291663

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂²×#12202

Subgroups

	subgroup	G/H
center, socle	C₂⁴	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁵
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

32005 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:

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

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#485039, #72292064, #72292073
maximal normal subgroup of	C₂²×#485039, #72292064, #72292073

GAP

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

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

C22×#12202 #72291663