C₂³×#2346 #72325377

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

Aliases

direct product

C₂³×#2346

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

101829 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)(11,12,14,15)(13,17,18,16)(19,20)(21,26)(22,25)(23,24);(7,9,10,8)(12,15)(13,18)(20,24)(21,25);(3,4);(5,6);(11,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,25)(24,26);(1,2)(3,4)(5,6);(11,13,14,18)(12,16,15,17)(19,22)(20,25)(21,24)(23,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#39862, C₂³×#35995, C₂³×#34415
maximal normal subgroup of	C₂³×#39862, C₂³×#35995, C₂³×#34415
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)(11,12,14,15)(13,17,18,16)(19,20)(21,26)(22,25)(23,24), (7,9,10,8)(12,15)(13,18)(20,24)(21,25), (3,4), (5,6), (11,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,25)(24,26), (1,2)(3,4)(5,6), (11,13,14,18)(12,16,15,17)(19,22)(20,25)(21,24)(23,26));

C23×#2346 #72325377