C₂×#7779623 #72324754

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

Aliases

direct product

C₂×#7779623

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 6734 conjugacy classes of subgroups; only the 31925 normal subgroups are shown.

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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232070
maximal normal subgroup of	C₂×#72232070
maximal quotient of	C₂×#72232136

GAP

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

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

C2×#7779623 #72324754