C₂×#7736723 #72330173

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#7736723

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

65413 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 26 points:

(3,4)(5,8)(6,10)(7,9)(11,12,14,16)(13,17,18,15)(19,20)(21,26)(22,25)(23,24);(3,5)(4,8)(6,9)(7,10)(11,12,14,16)(13,15,18,17)(20,24)(21,25);(3,4)(5,6)(7,9)(8,10)(11,12)(13,17)(14,16)(15,18)(19,21)(20,22)(23,25)(24,26);(11,12,14,16)(13,15,18,17)(19,22)(20,25)(21,24)(23,26);(1,2);(3,4)(5,8)(6,10)(7,9)(11,12)(13,17)(14,16)(15,18)(19,21)(20,22)(23,25)(24,26);(3,6,7,8)(4,5,9,10)(11,13,14,18)(12,15,16,17)(19,22)(20,25)(21,24)(23,26)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232277
maximal normal subgroup of	C₂×#72232277
maximal quotient of	C₂×#72323312

GAP

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

gap> G := Group((3,4)(5,8)(6,10)(7,9)(11,12,14,16)(13,17,18,15)(19,20)(21,26)(22,25)(23,24), (3,5)(4,8)(6,9)(7,10)(11,12,14,16)(13,15,18,17)(20,24)(21,25), (3,4)(5,6)(7,9)(8,10)(11,12)(13,17)(14,16)(15,18)(19,21)(20,22)(23,25)(24,26), (11,12,14,16)(13,15,18,17)(19,22)(20,25)(21,24)(23,26), (1,2), (3,4)(5,8)(6,10)(7,9)(11,12)(13,17)(14,16)(15,18)(19,21)(20,22)(23,25)(24,26), (3,6,7,8)(4,5,9,10)(11,13,14,18)(12,15,16,17)(19,22)(20,25)(21,24)(23,26));

C2×#7736723 #72330173