C₂²×#10937 #72316971

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂²×#10937

Subgroups

	subgroup	G/H
center	C₂³×C₄	#120
commutator	C₂³	C₂³×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂⁴	#386
prime radical	C₂⁷	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

30301 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)(16,21)(17,22)(23,24);(1,2);(6,7)(9,10,15,23,13,16,19,22)(11,14,20,24,18,21,12,17);(5,7,8,6)(10,16)(11,18)(15,19)(17,24);(3,4)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#364592, C₂×#72219669, C₂×#72238752
maximal normal subgroup of	C₂²×#364592, C₂×#72219669, C₂×#72238752

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

C22×#10937 #72316971