C₂³×#15267 #72342757

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²²×C₄

Aliases

direct product

C₂³×#15267

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3530120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#647670, C₂³×#648360
maximal normal subgroup of	C₂³×#647670, C₂³×#648360

GAP

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

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

C23×#15267 #72342757