C₂×#72224648 #72245298

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#72224648

Subgroups

	subgroup	G/H
center, socle	C₂²	#93877
commutator	C₃₂	C₂²×C₁₆
Frattini, cosocle	#7330	C₂³
prime radical	C₂³	#7457
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#1311 ◁ C₂×#7330 ◁ G
LECS	G ▷ #7330 ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

240 conjugacy classes of subgroups.

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72224662, D₃×#72224648, #72286639, …
maximal normal subgroup of	C₂×#72224662, #72286639, #72287062
maximal quotient of	D₃×#72224648, #72286639

GAP

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

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

C2×#72224648 #72245298