C₂²×#686390 #72321623

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹×C₄

Aliases

direct product

C₂²×#686390

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

891032 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)(11,12)(17,18,20,22)(19,21,23,24);(6,8)(10,11)(13,14,15,16)(19,23)(21,24);(6,8)(13,14)(15,16);(1,2)(3,4);(17,18)(19,24)(20,22)(21,23);(5,6,7,8)(9,11,12,10)(17,19,20,23)(18,21,22,24)

Conjugacy classes & character table

368 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×#72278357

GAP

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

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

C22×#686390 #72321623