C₂×#7854851 #72324353

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

Aliases

direct product

C₂×#7854851

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

At least 6918 conjugacy classes of subgroups; only the 31973 normal subgroups are shown.

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:

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

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232032, C₂×#72232070
maximal normal subgroup of	C₂×#72232032, C₂×#72232070
maximal quotient of	C₂×#72232060, C₂×#72232158

GAP

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

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

C2×#7854851 #72324353