C₂²×#7677103 #72339122

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂²×#7677103

Subgroups

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

1547224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 8 generators on 24 points:

(9,10)(11,12)(13,14)(15,16)(18,21)(19,23);(6,8)(9,11)(10,12)(13,16)(14,15)(18,21)(19,23);(6,8)(10,14)(12,15);(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24);(3,4);(1,2)(3,4);(9,11)(10,12)(13,16)(14,15);(5,6)(7,8)(9,12)(10,16)(11,14)(13,15)(17,19,20,23)(18,22,21,24)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72236233
maximal normal subgroup of	C₂²×#72236233

GAP

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

gap> G := Group((9,10)(11,12)(13,14)(15,16)(18,21)(19,23), (6,8)(9,11)(10,12)(13,16)(14,15)(18,21)(19,23), (6,8)(10,14)(12,15), (9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24), (3,4), (1,2)(3,4), (9,11)(10,12)(13,16)(14,15), (5,6)(7,8)(9,12)(10,16)(11,14)(13,15)(17,19,20,23)(18,22,21,24));

C22×#7677103 #72339122