C₂²×#17082 #72296850

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄²

Aliases

direct product

C₂²×#17082

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

80877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 24 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#5473828, C₂²×#2760366, C₂²×#4033889
maximal normal subgroup of	C₂²×#5473828, C₂²×#2760366, C₂²×#4033889

GAP

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

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

C22×#17082 #72296850