C₂×#7641846 #72255587

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

Aliases

direct product

C₂×#7641846

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

161717 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 64

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

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236365
maximal normal subgroup of	C₂×#72236365

GAP

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

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

C2×#7641846 #72255587