C₂×#135930 #72320935

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#135930

Subgroups

	subgroup	G/H
center, socle	C₂³	#1864
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂³×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1651 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 42 points:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72319899, C₂×#72319904, C₂×#72319964
maximal normal subgroup of	C₂×#72319899, C₂×#72319904, C₂×#72319964

GAP

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

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

C2×#135930 #72320935