C₂×#7681801 #72298756

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

Aliases

direct product

C₂×#7681801

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

120201 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297215
maximal normal subgroup of	C₂×#72297215
maximal quotient of	C₂×#72297195, C₂×#72298079

GAP

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

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

C2×#7681801 #72298756