C₂²×#36216 #72324873

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#36216

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#546	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

103101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72323403, C₂×#72323401, C₂²×#7659296
maximal normal subgroup of	C₂×#72323403, C₂×#72323401, C₂²×#7659296

GAP

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

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

C22×#36216 #72324873