C₂×#94506 #72245258

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₈² · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#94506

Subgroups

	subgroup	G/H
center	C₂×C₈	C₈²
commutator	C₈	C₂×C₈²
Frattini, cosocle	#1397	C₂³
socle	C₂²	#7330
prime radical	C₂³	#1398
LCS	G ▷ C₈ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ #1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

225 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 181 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);(3,4,5,11,6,13,19,35,7,15,20,33,23,41,49,58,8,16,21,34,24,42,48,59,25,10,28,29,51,30,53,55,9,14,18,36,22,40,46,61,26,32,47,54,50,57,66,60,27,12,17,37,43,38,44,62,52,31,45,56,65,39,63,64);(4,10,12,15,14,31,16,32)(5,17,18,21)(6,22)(11,33,34,29,36,54,37,56)(13,39,38,57,40,30,42,41)(19,48,46,44)(20,45,47,28)(23,50)(24,43)(35,60,59,64,61,58,62,55)(49,53,66,63)(51,65)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72224644, C₂×#72224648, C₂×#72224652, …
maximal normal subgroup of	C₂×#72224644, C₂×#72224648, C₂×#72224652, …
maximal quotient of	D₃×#94506, #72286400

GAP

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

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