C₂²×#35729 #72325838

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

Aliases

direct product

C₂²×#35729

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

106773 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7849737, C₂²×#7631422
maximal normal subgroup of	C₂²×#7849737, C₂²×#7631422

GAP

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

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

C22×#35729 #72325838