C₂×#243908 #72284510

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#243908

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂³	C₂×#399
prime radical	C₂×#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

4719 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 42 points:

(1,2);(4,7)(5,9)(8,10)(11,12,15,19,16,20,27,34)(13,17,22,29,23,30,37,41)(14,18,25,32,26,33,40,42)(21,35,24,38,36,28,39,31);(4,8)(7,10);(11,13)(12,17)(14,24,26,39)(15,22)(16,23)(18,31,33,38)(19,29)(20,30)(21,25,36,40)(27,37)(28,32,35,42)(34,41);(3,4,9,10,6,8,5,7)(11,14)(12,18)(13,21,23,36)(15,25)(16,26)(17,28,30,35)(19,32)(20,33)(22,24,37,39)(27,40)(29,31,41,38)(34,42)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72283833
maximal normal subgroup of	C₂×#72283833

GAP

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

gap> G := Group((1,2), (4,7)(5,9)(8,10)(11,12,15,19,16,20,27,34)(13,17,22,29,23,30,37,41)(14,18,25,32,26,33,40,42)(21,35,24,38,36,28,39,31), (4,8)(7,10), (11,13)(12,17)(14,24,26,39)(15,22)(16,23)(18,31,33,38)(19,29)(20,30)(21,25,36,40)(27,37)(28,32,35,42)(34,41), (3,4,9,10,6,8,5,7)(11,14)(12,18)(13,21,23,36)(15,25)(16,26)(17,28,30,35)(19,32)(20,33)(22,24,37,39)(27,40)(29,31,41,38)(34,42));

C2×#243908 #72284510