C₂×#7653764 #72256251

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂×#7653764

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

114415 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232033, C₂²×#7653764
maximal normal subgroup of	C₂×#72232033, C₂²×#7653764
maximal quotient of	C₂×#72233571, C₂×#72233567, C₂×#72297599, …

GAP

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

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

C2×#7653764 #72256251