C₂×#72278194 #72296469

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶×C₄

Aliases

direct product

C₂×#72278194

Subgroups

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

At least 9188 conjugacy classes of subgroups; only the 8288 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

272 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#72278229
maximal normal subgroup of	C₂×#72278229

GAP

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

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

C2×#72278194 #72296469