C₂³×#3488 #72278218

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁷

Aliases

direct product

C₂³×#3488

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

884421 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 8 generators on 18 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#7684300, C₂³×D₄×H(5,𝔽₂), C₂³×#62892, …
maximal normal subgroup of	C₂²×#7684300, C₂³×D₄×H(5,𝔽₂), C₂³×#62892, …
maximal quotient of	C₂³×#35680, C₂²×#7684300, C₂³×D₄×#121, …

GAP

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

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

C23×#3488 #72278218