C₂×#128050 #72314540

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄³

Aliases

direct product

C₂×#128050

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#342
commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	C₂⁴×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂³×C₄×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

5697 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:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72238685, C₂×#72234076
maximal normal subgroup of	C₂×#72238685, C₂×#72234076

GAP

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

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

C2×#128050 #72314540