C₂²×#16501 #72337028

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂²×#16501

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

79893 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#846769, C₂²×#759718, C₂²×#846737, …
maximal normal subgroup of	C₂²×#846769, C₂²×#759718, C₂²×#846737, …

GAP

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

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

C22×#16501 #72337028