C₂²×#37840 #72324716

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#37840

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

111289 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7711159, C₂²×#7716499, C₂²×#7716427, …
maximal normal subgroup of	C₂²×#7711159, C₂²×#7716499, C₂²×#7716427, …
maximal quotient of	C₂²×#652786, C₂²×#652806, C₂²×#846577

GAP

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

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

C22×#37840 #72324716