C₂²×#7659296 #72333434

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂²×#7659296

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 6141 conjugacy classes of subgroups; only the 446936 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 8 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72323423
maximal normal subgroup of	C₂×#72323423

GAP

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

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

C22×#7659296 #72333434