C₂×#93877 #72231920

order	1024
nilpotency class	5
isoclinism	isoclinic class of #7457
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#93877

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#1368
commutator	C₁₆	C₂²×C₁₆
Frattini, cosocle	#1311	C₂³
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#1311 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

221 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 221 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 3 generators on 50 points:

(1,2);(3,4,5,8,6,9,11,14,7,10,12,15,13,16,17,18)(20,25,26,35,27,36,39,38)(21,29,30,43)(22,32)(23,34)(28,42,40,37,41,49,48,50)(31,46,45,44)(33,47);(19,20,21,25,22,28,29,37,23,26,31,38,32,40,44,42,24,27,30,36,33,41,43,50,34,39,45,35,47,48,46,49)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#93877, #72231930, #72231933, …
maximal normal subgroup of	C₂²×#93877, #72231930, #72231933, …
maximal quotient of	C₂²×#93877, D₃×#93877, C₂×#72224648, …

GAP

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

gap> G := Group((1,2), (3,4,5,8,6,9,11,14,7,10,12,15,13,16,17,18)(20,25,26,35,27,36,39,38)(21,29,30,43)(22,32)(23,34)(28,42,40,37,41,49,48,50)(31,46,45,44)(33,47), (19,20,21,25,22,28,29,37,23,26,31,38,32,40,44,42,24,27,30,36,33,41,43,50,34,39,45,35,47,48,46,49));