C₂³×#2360 #72339281

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂³×#2360

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁷	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

146549 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 22 points:

(11,12,14,13)(15,16)(17,20)(18,21)(19,22);(11,12,14,13)(15,17)(16,19)(18,22)(20,21);(3,4);(5,6);(8,10)(15,17)(16,20)(18,22)(19,21);(1,2)(3,4)(5,6);(7,8,9,10)(11,13,14,12)(15,16)(17,19)(18,21)(20,22)

Conjugacy classes & character table

400 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂³×#36107, C₂³×C₄×#546, C₂³×#36441, …
maximal normal subgroup of	C₂³×#36107, C₂³×C₄×#546, C₂³×#36441, …

GAP

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

gap> G := Group((11,12,14,13)(15,16)(17,20)(18,21)(19,22), (11,12,14,13)(15,17)(16,19)(18,22)(20,21), (3,4), (5,6), (8,10)(15,17)(16,20)(18,22)(19,21), (1,2)(3,4)(5,6), (7,8,9,10)(11,13,14,12)(15,16)(17,19)(18,21)(20,22));

C23×#2360 #72339281