C₂×#1419353 #72294745

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#1419353

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

53559 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293914
maximal normal subgroup of	C₂×#72293914
maximal quotient of	C₂×#72293928, C₂×#72293943

GAP

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

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

C2×#1419353 #72294745