C₂³×#1914 #72342408

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂³×#1914

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#386
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂²×C₄	C₂⁶
prime radical	C₂³×#535	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

68261 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#6437988
maximal normal subgroup of	C₂²×#6437988

GAP

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

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

C23×#1914 #72342408