C₂²×#12090 #72291664

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂²×#12090

Subgroups

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

34569 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)(7,9)(8,11)(10,12)(15,19);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(14,17)(18,20);(1,2);(3,4);(1,2)(3,4)(6,9)(7,12)(10,11)(13,14,16,18)(15,17,19,20)

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#485039, #72292059, #72292068
maximal normal subgroup of	C₂²×#485039, #72292059, #72292068

GAP

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

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

C22×#12090 #72291664