C₂³×#2379 #72325380

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂¹⁷×C₄²

Aliases

direct product

C₂³×#2379

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

112485 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)(13,15)(14,16)(17,18)(19,20,23,24)(21,22,26,25);(7,8,10,9)(12,13)(16,17)(20,24)(22,25);(3,4);(5,6);(7,9,10,8)(11,13,15,12)(14,16,18,17)(19,21)(20,25)(22,24)(23,26);(1,2)(3,4)(5,6);(11,14)(12,16)(13,17)(15,18)(19,22,23,25)(20,21,24,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#50864, C₂³×#34415, C₂³×#34406
maximal normal subgroup of	C₂³×#50864, C₂³×#34415, C₂³×#34406
maximal quotient of	C₂³×#14473, C₂³×#14662

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

C23×#2379 #72325380