C₂²×#11100 #72316989

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂²×#11100

Subgroups

	subgroup	G/H
center	C₂³×C₄	#127
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂⁴	#392
prime radical	C₂⁷	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

29701 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)(16,24)(17,22)(18,23);(1,2);(5,6)(7,8)(9,10,15,16,13,17,20,23)(11,14,21,24,19,22,12,18);(6,8)(10,16,17,23)(11,19)(12,21)(14,18,22,24);(3,4)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#364592, C₂×#72239546, C₂×#72237215
maximal normal subgroup of	C₂²×#364592, C₂×#72239546, C₂×#72237215

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

C22×#11100 #72316989