C₂×#7737454 #72334564

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

Aliases

direct product

C₂×#7737454

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

109809 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236233, #72337228, C₂²×#7737454
maximal normal subgroup of	C₂×#72236233, #72337228, C₂²×#7737454
maximal quotient of	C₂×#72334338, C₂×#72334339, C₂×#72334340, …

GAP

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

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

C2×#7737454 #72334564