C₂×#7737421 #72298993

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

Aliases

direct product

C₂×#7737421

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

54557 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297217
maximal normal subgroup of	C₂×#72297217
maximal quotient of	C₂×#72297597

GAP

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

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

C2×#7737421 #72298993