C₂×#72278103 #72296480

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁷

Aliases

direct product

C₂×#72278103

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂⁴	C₂⁵×C₄
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂³×#2404	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 7946 conjugacy classes of subgroups; only the 12912 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);(4,6)(8,9)(11,12,13,14)(15,16)(17,18);(4,6)(7,8,10,9)(11,12)(13,14)(19,20,22,24)(21,23,25,26);(4,6)(8,9)(11,12,13,14)(19,20)(21,23)(22,24)(25,26);(3,4,5,6)(7,9,10,8)(11,12)(13,14)(19,21,22,25)(20,23,24,26);(1,2)

Conjugacy classes & character table

368 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), (4,6)(8,9)(11,12,13,14)(15,16)(17,18), (4,6)(7,8,10,9)(11,12)(13,14)(19,20,22,24)(21,23,25,26), (4,6)(8,9)(11,12,13,14)(19,20)(21,23)(22,24)(25,26), (3,4,5,6)(7,9,10,8)(11,12)(13,14)(19,21,22,25)(20,23,24,26), (1,2));

C2×#72278103 #72296480