C₂²×#11073 #72316863

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂²×#11073

Subgroups

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

29181 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)(12,20,19,22)(13,14,18,23)(15,21,17,24);(1,2);(5,6,7,8)(10,14)(11,18)(17,20)(19,24);(5,6)(7,8)(10,15)(12,21)(14,22)(16,17)(19,24)(20,23);(3,4)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#394249, C₂×#72239552, C₂×#72239574
maximal normal subgroup of	C₂²×#394249, C₂×#72239552, C₂×#72239574

GAP

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

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

C22×#11073 #72316863