C₂×#7708709 #72339969

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

Aliases

direct product

C₂×#7708709

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

108649 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)(7,8,11,12)(9,10,13,14)(16,20)(18,21);(15,16,19,20)(17,21,22,18);(8,12)(10,14)(15,17)(16,21)(18,20)(19,22);(4,6);(7,9)(8,10)(11,13)(12,14)(15,17)(16,21)(18,20)(19,22);(3,4)(5,6)(7,10)(8,13)(9,12)(11,14)(15,18,19,21)(16,22,20,17);(1,2)

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×#72337223, C₂×#72337327

GAP

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

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

C2×#7708709 #72339969