C₂×#7685423 #72304252

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

Aliases

direct product

C₂×#7685423

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 10302 conjugacy classes of subgroups; only the 32189 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:

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

Conjugacy classes & character table

226 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72234464
maximal normal subgroup of	C₂×#72234464

GAP

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

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

C2×#7685423 #72304252