C₂×#94509 #72245265

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#94509

Subgroups

	subgroup	G/H
center	C₂×C₈	#322
commutator	C₈	C₂×C₄×C₁₆
Frattini, cosocle	#1397	C₂³
socle	C₂²	#7332
prime radical	C₂³	#1398
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#1397 ◁ G
LECS	G ▷ #1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

217 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 165 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) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 66 points:

(1,2);(3,4,6,12,8,14,26,39,9,16,27,41,25,52,62,65,10,17,28,42,35,49,58,38,36,54,5,19,20,47,21,56,11,18,22,43,7,29,30,50,31,15,23,40,32,51,60,55,37,53,61,66,33,13,44,45,63,46,24,57,59,48,34,64);(3,5,7,21,9,22,32,30,10,23,33,60,36,61,59,44,11,24,8,34,31,6,25,26,37,27,35,62,63,28,20,58)(4,12,13,38,15,40,47,64,17,42,14,50,46,57,51,65,18,43,49,45,16,41,48,56,53,66,29,39,54,19,52,55)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72224644, C₂×#72224654, C₂×#72224659, …
maximal normal subgroup of	C₂×#72224644, C₂×#72224654, C₂×#72224659, …
maximal quotient of	D₃×#94509, #72286426

GAP

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

gap> G := Group((1,2), (3,4,6,12,8,14,26,39,9,16,27,41,25,52,62,65,10,17,28,42,35,49,58,38,36,54,5,19,20,47,21,56,11,18,22,43,7,29,30,50,31,15,23,40,32,51,60,55,37,53,61,66,33,13,44,45,63,46,24,57,59,48,34,64), (3,5,7,21,9,22,32,30,10,23,33,60,36,61,59,44,11,24,8,34,31,6,25,26,37,27,35,62,63,28,20,58)(4,12,13,38,15,40,47,64,17,42,14,50,46,57,51,65,18,43,49,45,16,41,48,56,53,66,29,39,54,19,52,55));