C₂×#8001029 #72331548

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#8001029

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

83233 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 22 points:

(3,4)(5,6)(15,16)(17,20)(18,19)(21,22);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(16,18)(20,21);(8,11)(12,13)(15,17)(16,20)(18,21)(19,22);(4,5)(7,8)(9,12)(10,11)(13,14)(15,16,19,18)(17,21,22,20);(1,2);(3,4,6,5)(7,8)(9,13)(10,11)(12,14)(15,18,19,16)(17,20,22,21);(3,5,6,4)(7,9,10,14)(8,12,11,13)(15,17)(16,20)(18,21)(19,22)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232293
maximal normal subgroup of	C₂×#72232293
maximal quotient of	C₂×#72323402

GAP

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

gap> G := Group((3,4)(5,6)(15,16)(17,20)(18,19)(21,22), (3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(16,18)(20,21), (8,11)(12,13)(15,17)(16,20)(18,21)(19,22), (4,5)(7,8)(9,12)(10,11)(13,14)(15,16,19,18)(17,21,22,20), (1,2), (3,4,6,5)(7,8)(9,13)(10,11)(12,14)(15,18,19,16)(17,20,22,21), (3,5,6,4)(7,9,10,14)(8,12,11,13)(15,17)(16,20)(18,21)(19,22));

C2×#8001029 #72331548