C₂×#7722014 #72332275

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

Aliases

direct product

C₂×#7722014

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

93469 conjugacy classes of subgroups.

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323293
maximal normal subgroup of	C₂×#72323293
maximal quotient of	C₂×#72323406

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

C2×#7722014 #72332275