C₂×#7673820 #72332289

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

Aliases

direct product

C₂×#7673820

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

125189 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323293
maximal normal subgroup of	C₂×#72323293
maximal quotient of	C₂×#72240162

GAP

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

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

C2×#7673820 #72332289