C₂×#8001030 #72331584

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

Aliases

direct product

C₂×#8001030

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 10604 conjugacy classes of subgroups; only the 31821 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232293
maximal normal subgroup of	C₂×#72232293
maximal quotient of	C₂×#72263089

GAP

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

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

C2×#8001030 #72331584