C₂×#7670332 #72341972

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

Aliases

direct product

C₂×#7670332

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

66893 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)(5,6)(7,10)(8,9)(20,22)(23,25);(3,4,5,6)(7,8,9,10)(11,12)(13,15)(14,17)(16,18)(19,20,21,22)(23,26,25,24);(1,2)(3,5)(4,6)(11,13,16,17)(12,15,18,14)(19,21)(20,22)(23,25)(24,26);(3,4,5,6)(7,8,9,10)(11,14,16,15)(12,17,18,13)(19,20,21,22)(23,24,25,26);(4,6)(7,9)(19,20,21,22)(23,24,25,26);(3,6,5,4)(7,8,9,10)(11,12)(13,15)(14,17)(16,18)(19,22,21,20)(23,26,25,24);(3,7)(4,8)(5,9)(6,10)(11,15,16,14)(12,17,18,13)(19,23,21,25)(20,24,22,26)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72241249
maximal normal subgroup of	C₂×#72241249

GAP

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

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

C2×#7670332 #72341972