C₂²×#10948 #72316955

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

direct product

C₂²×#10948

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#386
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

30429 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#364592, C₂×#72226728, C₂×#72239574
maximal normal subgroup of	C₂²×#364592, C₂×#72226728, C₂×#72239574

GAP

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

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

C22×#10948 #72316955