C₂×#7677037 #72323797

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

Aliases

direct product

C₂×#7677037

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

93873 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)(9,13)(12,14);(3,4)(5,6)(15,16,19,18)(17,20,21,22);(9,13)(12,14)(15,17)(16,20)(18,22)(19,21);(1,2);(8,11)(12,14);(7,8,10,11)(9,12,13,14)(15,18,19,16)(17,20,21,22);(7,9,10,13)(8,12,11,14)

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72238224
maximal normal subgroup of	C₂×#72238224
maximal quotient of	C₂×#72323274, C₂×#72323422

GAP

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

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

C2×#7677037 #72323797