C₂²×#34222 #72299862

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#34222

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

94717 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#7746558
maximal normal subgroup of	C₂²×#7746558

GAP

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

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

C22×#34222 #72299862