C₂×#7631308 #72304233

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

Aliases

direct product

C₂×#7631308

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 6156 conjugacy classes of subgroups; only the 32613 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:

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

Conjugacy classes & character table

244 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72234467
maximal normal subgroup of	C₂×#72234467

GAP

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

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

C2×#7631308 #72304233