C₂×#143163 #72283239

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#143163

Subgroups

	subgroup	G/H
center, socle	C₂²	#10018
commutator, Frattini, cosocle	C₄×M₁₆	C₂⁴
prime radical	C₂×#33688	C₂
LCS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2129 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:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72231251
maximal normal subgroup of	C₂×#72231251

GAP

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

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

C2×#143163 #72283239