C₂²×#517627 #72321628

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂²×#517627

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁴×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

776504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72279977
maximal normal subgroup of	C₂²×#72279977
maximal quotient of	C₂²×C₂²⋊C₄×#392

GAP

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

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

C22×#517627 #72321628