C₂×#8075294 #72338162

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

Aliases

direct product

C₂×#8075294

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

87169 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);(3,4)(5,6)(8,11)(9,13)(17,20)(18,22);(9,13)(12,14)(15,16)(17,18)(19,21)(20,22);(4,6)(7,8)(9,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,22);(3,4)(5,6)(15,18)(16,20)(17,21)(19,22);(7,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,22)(20,21);(7,9)(8,12)(10,13)(11,14)(15,16)(17,22)(18,20)(19,21)

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#72340506, C₂×#72341141
maximal subgroup of	C₂×#72312792, C₂×#72238249, C₂×#72340506
maximal normal subgroup of	C₂×#72312792, C₂×#72238249, C₂×#72340506

GAP

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

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

C2×#8075294 #72338162