C₂³×#10586472 #72301685

order	4096
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂¹⁰ · Schur multiplier (H₂) = C₂⁴³

Aliases

direct product

C₂³×#10586472

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁷
commutator, Frattini, cosocle	C₂²	C₂¹⁰
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

258661611 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 10 generators on 22 points:

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

Conjugacy classes & character table

1168 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

C₂³×#72236189

GAP

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

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

C23×#10586472 #72301685