C₂³×#59123 #72289494

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁷

Aliases

direct product

C₂³×#59123

Subgroups

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

2353248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#7684106, C₂³×#7918389
maximal normal subgroup of	C₂³×#7684106, C₂³×#7918389
maximal quotient of	C₂³×#1418391

GAP

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

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

C23×#59123 #72289494