C₂×#8063770 #72299863

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

Aliases

direct product

C₂×#8063770

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 6382 conjugacy classes of subgroups; only the 31853 normal subgroups are shown.

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

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297207
maximal normal subgroup of	C₂×#72297207

GAP

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

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

C2×#8063770 #72299863