C₂²×#10776 #72316825

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂²×#10776

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#379
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

47117 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#394249, C₂×#72239564, C₂²×#364592, …
maximal normal subgroup of	C₂²×#394249, C₂×#72239564, C₂²×#364592, …

GAP

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

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

C22×#10776 #72316825