C₂³×#14820 #72325667

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²²

Aliases

direct product

C₂³×#14820

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁹	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2930904 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#846041, C₂³×#1420724
maximal normal subgroup of	C₂³×#846041, C₂³×#1420724

GAP

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

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

C23×#14820 #72325667