C₂×#7631295 #72304238

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

Aliases

direct product

C₂×#7631295

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

94485 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:

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

Conjugacy classes & character table

244 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72234467, C₂×#72232067
maximal normal subgroup of	C₂×#72234467, C₂×#72232067
maximal quotient of	C₂×#72232187

GAP

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

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

C2×#7631295 #72304238