C₂²×#34437 #72324861

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

Aliases

direct product

C₂²×#34437

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#534	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

102749 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)(11,13)(12,15)(14,16)(17,18)(19,21)(20,22)(23,24);(9,11)(10,13)(12,16)(14,15)(17,18,20,22)(19,23,24,21);(9,10,12,15)(11,14,16,13)(17,19)(18,21)(20,24)(22,23);(3,4);(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,22)(21,24);(1,2)(3,4);(5,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,21)(20,24)(22,23)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#7763623
maximal normal subgroup of	C₂²×#7763623
maximal quotient of	C₂²×#540671

GAP

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

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

C22×#34437 #72324861