C₂²×#11157 #72342389

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

Aliases

direct product

C₂²×#11157

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#397
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

26845 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)(12,20)(16,17)(19,24)(21,23);(6,8)(9,10)(11,15,19,22)(12,17)(13,16)(14,20)(18,23,24,21);(1,2);(9,11,12,18,13,19,20,24)(10,15,14,21,16,22,17,23);(3,4)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#368859, C₂²×#394600
maximal normal subgroup of	C₂²×#368859, C₂²×#394600

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

C22×#11157 #72342389