C₂²×#35349 #72325840

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

Aliases

direct product

C₂²×#35349

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

107981 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7675441, C₂²×#7628711
maximal normal subgroup of	C₂²×#7675441, C₂²×#7628711

GAP

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

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

C22×#35349 #72325840