C₂³×#2343 #72325466

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂³×#2343

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁷	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

114933 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#38366, C₂³×#35683, C₂³×#34406
maximal normal subgroup of	C₂³×#38366, C₂³×#35683, C₂³×#34406
maximal quotient of	C₂³×#13957

GAP

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

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

C23×#2343 #72325466