C₂³×#45355 #72260642

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁸

Aliases

direct product

C₂³×#45355

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

2745064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72233929, #72261869, #72262133, …
maximal normal subgroup of	C₂×#72233929, #72261869, #72262133, …
maximal quotient of	C₂²×#72232559, C₂³×#1259281, C₂×#72327629
automorphism group of	#9279, #9280, #9281, …

GAP

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

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

C23×#45355 #72260642