C₂×#7850314 #72324784

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

Aliases

direct product

C₂×#7850314

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 10632 conjugacy classes of subgroups; only the 31933 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:

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

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232068, C₂×#72232293
maximal normal subgroup of	C₂×#72232068, C₂×#72232293
maximal quotient of	C₂×#72232156, C₂×#72323410

GAP

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

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

C2×#7850314 #72324784