C₂×#93878 #72245279

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#93878

Subgroups

	subgroup	G/H
center	C₂×C₄	#1368
commutator	C₁₆	C₂²×C₁₆
Frattini, cosocle	#1311	C₂³
socle	C₂²	#7099
prime radical	C₂³	#1369
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ C₂×#1311 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

201 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 125 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);(3,4,6,14,8,16,28,46,10,17,29,15,9,38,7,32,12,18,31,47,37,49,33,62,11,39,30,48,36,45,34,63)(5,19,20,40,22,50,21,42,24,52,27,61,59,66,57,43,26,53,56,41,35,51,58,65,25,44,55,64,23,60,54,13);(3,5,7,21,9,23,30,20,11,25,28,54,8,35,31,55,12,26,34,58,36,59,29,56,10,24,33,57,37,22,6,27)(4,13,15,19,16,41,32,51,17,42,47,52,38,64,62,60,18,43,48,53,49,40,63,50,39,65,14,44,45,61,46,66)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72224647, C₂×#72224648, C₂×#72224659, …
maximal normal subgroup of	C₂×#72224647, C₂×#72224648, C₂×#72224659, …
maximal quotient of	D₃×#93878, #72286621

GAP

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

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