C₂×#130290 #72280549

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#130290

Subgroups

	subgroup	G/H
center, socle	C₂²	#8350
commutator	C₂×M₁₆	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
prime radical	C₂×#62283	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

4198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

(4,10)(5,16)(7,25)(11,14)(12,28)(13,30)(15,29)(17,32)(18,20)(19,21)(26,31)(33,34);(3,4,8,13,9,14,27,15)(5,17,19,31,16,28,21,34)(6,11,22,30,24,10,23,29)(7,26,18,32,25,33,20,12);(3,5,9,16)(4,11)(6,20,24,18)(7,22,25,23)(8,21,27,19)(10,14)(12,31)(13,30)(15,29)(17,33)(26,28)(32,34);(1,2)

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280627, #72280655, C₂×#72231700, …
maximal normal subgroup of	#72280627, #72280655, C₂×#72231700, …

GAP

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

gap> G := Group((4,10)(5,16)(7,25)(11,14)(12,28)(13,30)(15,29)(17,32)(18,20)(19,21)(26,31)(33,34), (3,4,8,13,9,14,27,15)(5,17,19,31,16,28,21,34)(6,11,22,30,24,10,23,29)(7,26,18,32,25,33,20,12), (3,5,9,16)(4,11)(6,20,24,18)(7,22,25,23)(8,21,27,19)(10,14)(12,31)(13,30)(15,29)(17,33)(26,28)(32,34), (1,2));

C2×#130290 #72280549