C₂×#7678727 #72331669

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#7678727

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

123497 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)(8,10)(11,12)(13,20)(14,18)(15,19)(16,17)(21,22)(23,25)(24,26);(4,6)(7,8)(9,10)(11,13)(12,15)(14,21)(16,19)(17,20)(18,24)(22,25)(23,26);(3,4,5,6)(12,17)(13,19)(18,25)(21,26);(1,2);(3,4)(5,6)(11,14)(12,18)(13,21)(15,24)(16,23)(17,25)(19,26)(20,22);(11,12)(13,20)(14,18)(15,19)(16,17)(21,22)(23,25)(24,26);(11,15,16,20)(12,19,17,13)(14,22,23,24)(18,21,25,26)

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72327577
maximal normal subgroup of	C₂×#72327577
maximal quotient of	C₂×#72257916, C₂×#72327618

GAP

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

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

C2×#7678727 #72331669