C₂×#7711906 #72299005

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂×#7711906

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

93653 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297217
maximal normal subgroup of	C₂×#72297217
maximal quotient of	C₂×#72297596

GAP

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

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

C2×#7711906 #72299005