C₂×#7759725 #72298442

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

Aliases

direct product

C₂×#7759725

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

102697 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72234453, C₂×#72260861
maximal normal subgroup of	C₂×#72234453, C₂×#72260861
maximal quotient of	C₂×#72297397, C₂×#72298078

GAP

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

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

C2×#7759725 #72298442