C₂³×#2853 #72339231

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂³×#2853

Subgroups

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

133197 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#37232, C₂³×#36316, C₂³×#36621
maximal normal subgroup of	C₂³×#37232, C₂³×#36316, C₂³×#36621

GAP

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

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

C23×#2853 #72339231