C₂×#7733672 #72339958

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

Aliases

direct product

C₂×#7733672

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

109061 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339970
maximal normal subgroup of	#72339970
maximal quotient of	C₂×#72337224, #72339970, C₂×#72329230

GAP

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

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

C2×#7733672 #72339958